An optimal eighth-order class of three-step weighted

International Journal of Computer Mathematics

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An optimal eighth-order class of three-step weighted Newton's methods and their dynamics behind the purely imaginary extraneous fixed points Min Surp Rhee, Young Ik Kim & Beny Neta To cite this article: Min Surp Rhee, Young Ik Kim & Beny Neta (2017): An optimal eighthorder class of three-step weighted Newton's methods and their dynamics behind the purely imaginary extraneous fixed points, International Journal of Computer Mathematics, DOI: 10.1080/00207160.2017.1367387 To link to this article: http://dx.doi.org/10.1080/00207160.2017.1367387

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Date: 30 August 2017, At: 04:20

INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS, 2017 https://doi.org/10.1080/00207160.2017.1367387

ARTICLE

An optimal eighth-order class of three-step weighted Newton’s methods and their dynamics behind the purely imaginary extraneous fixed points Min Surp Rheea , Young Ik Kima and Beny Netab

Downloaded by [Young Ik Kim] at 04:20 30 August 2017

a Department of Applied Mathematics, Dankook University, Cheonan, Republic of Korea; b Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA, USA

ABSTRACT

ARTICLE HISTORY

In this paper, we not only develop an optimal class of three-step eighthorder methods with higher order weight functions employed in the second and third sub-steps, but also investigate their dynamics underlying the purely imaginary extraneous fixed points. Their theoretical and computational properties are fully described along with a main theorem stating the order of convergence and the asymptotic error constant as well as extensive studies of special cases with rational weight functions. A number of numerical examples are illustrated to confirm the underlying theoretical development. Besides, to show the convergence behaviour of global character, fully explored is the dynamics of the proposed family of eighthorder methods as well as an existing competitive method with the help of illustrative basins of attraction.

Received 1 February 2017 Revised 25 May 2017 Accepted 17 July 2017 KEYWORDS

Eighth-order convergence; weight function; asymptotic error constant; efficiency index; purely imaginary extraneous fixed point; basin of attraction 2000 AMS SUBJECT CLASSIFICATIONS

65H05; 65H99

1. Introduction Nonlinear equations of high complexity naturally arise when describing our daily-life physical phenomena such as the evolving dynamics of a spinning tennis ball, a swinging pendulum, violent whirling windstorms, turbulent fluid flow as well as unpredictable weather forecast. Since exact solutions are rarely available, we usually resort to the classical second-order Newton’s method for the numerical solutions. Since Traub [40] made a pioneering work in the 1960s toward the qualitative and quantitative analyses of iterative methods locating numerical roots for nonlinear equations, many authors [7,14,17,18,21,23,26,34,37] have developed high-order multipoint methods. Petković et al. [33] collected and updated the state of the art of multipoint methods. A numerical scheme is said to be optimal according to Kung–Traub’s conjecture [24] that any multipoint method without memory can attain its convergence order of at most 2k−1 for k functional evaluations with k ∈ N. For the sake of comparison, we first introduce an existing eighth-order method in [37] presented by Equation (1). • Sharma-Arora method (SA) yn = xn −

f (xn ) , f (xn )

zn = yn −

f (yn ) = xn − 2f [yn , xn ] − f (xn )

CONTACT Young Ik Kim 330-714, Republic of Korea

[email protected]

© 2017 Informa UK Limited, trading as Taylor & Francis Group

1−s 1 − 2s

·

f (xn ) , f (xn )

Department of Applied Mathematics, Dankook University, Cheonan

2

M. S. RHEE ET AL.

f [zn , yn ] f (zn ) · f [zn , xn ] 2f [zn , yn ] − f [zn , xn ] 1−s su(1 − s)(1 − u) f (xn ) = xn − 1+ · , 1 − 2s (1 − su)(1 − 2s − 2u + 3su) f (xn )

xn+1 = zn −

(1)

where f [r, t] = (f (r) − f (t))/(r − t), s = f (yn )/f (xn ) and u = f (zn )/f (yn ).


Method (1) has been found to be very competitive judging from the recent studies performed by Lee et al. [25] and Chun and Neta [11], which motivates us to develop a new class of efficient methods. In this paper, we shall seek a class of optimal eighth-order simple-root finders that are competitive against or comparable to method (1). To this end, we employ an optimal three-step high-order family of iterative methods in the form of weighted Newton-like simple-root finders below: yn = xn −

f (xn ) , f (xn )

zn = xn − Lf (s) ·

f (xn ) , f (xn )

xn+1 = zn − Kf (s, u) ·

(2)

f (xn ) f (xn ) = xn − [Lf (s) + Kf (s, u)] · , f (xn ) f (xn )

where s and u are given in Equation (1) and Lf : C → C is a weight function being analytic [1] in a neighbourhood of 0 and Kf : C2 → C is a weight function being holomorphic [22,36] in a neighbourhood of (0, 0). Note that Equation (1) is a special case of Equation (2) with Lf (s) = (1 − s)/(1 − 2s) and Kf (s, u) = su(1 − s)2 (1 − u)/{(1 − 2s)(1 − su)(1 − 2s − 2u + 3su)}. It is interesting to see that Equation (1) can be expressed by means of fifth-order rational weight function Kf (s, u) without using divided differences. The forms of Equation (2) use three functional values plus a single derivative without using divided differences as used in Equation (1). Definition 1.1 (Error equation, asymptotic error constant, order of convergence): Let x0 , x1 , . . . , xn , . . . be a sequence of numbers converging to α. Let en = xn − α for n = 0, 1, 2, . . .. If constants p+1 p ≥ 1, c = 0 exist in such a way that en+1 = c en p + O(en ) called the error equation, then p and η = |c| are said to be the order of convergence and the asymptotic error constant, respectively. It is easy to find c = limn→∞ en+1 /en p . Some authors call c itself the asymptotic error constant. In this paper, we aim not only to design a class of optimal eighth-order methods by fully specifying the algebraic structure of generic weight functions Lf (s) and Kf (s, u), but also to investigate their dynamics by means of basins of attractions [16] (to be discussed in Section 5) behind the purely imaginary extraneous fixed points [42] (to be described in Section 4) when applied to a prototype quadratic polynomial. The last sub-step of Equation (2) in the form of weighted Newton’s method is clearly more convenient in dealing with extraneous fixed points that can be found directly from the roots of the weight function Lf (s) + Kf (s, u). It is of importance for us to pursue suitable parameters giving the basin of attraction with a larger region of convergence. The presence of extraneous fixed points may induce attractive, indifferent, repulsive as well as other chaotic orbits influencing the relevant dynamics of the iterative methods. Notice that the imaginary axis symmetrically divides the whole complex plane into two half planes. Since we observe the convergence behaviour in the dynamical planes through the basins of attraction in the form of a square region centred at the origin, the resulting dynamics behind the extraneous fixed points on the symmetry (imaginary) axis is expected to be less influenced by the presence of the possible periodic or chaotic attractors. Thus, in the current analysis, it would be preferable for

INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS

3

us to choose free parameters in such a way that the extraneous fixed points should be located on the imaginary axis. In Section 2, the main theorem regarding the convergence behaviour is described with appropriate forms of two weight functions Lf and Kf . Section 3 investigates some special cases of Kf (s, u). Section 4 discusses the purely imaginary extraneous fixed points together with their stabilities and investigates their theoretical multipliers. Section 5 presents numerical experiments along with the illustration of the relevant dynamics and describes concluding remarks at the end.

2. Main theorem


We shall state in this section the main theorem with generic weight functions Lf (s) and Kf (s, u) employed: Theorem 2.1: Assume that f : C → C has a simple root α and is analytic in a region containing α. Let cj = f (j) (α)/j!f (α) for j = 2, 3, . . .. Let x0 be an initial guess chosen in a sufficiently small neighbourhood of α. Let Lf : C → C be analytic in a neighbourhood of 0. Let Li = (1/i!)(di /dsi )Lf (s)|(s=0) for 0 ≤ i ≤ 7. Let Kf : C2 → C be holomorphic in a neighbourhood of (0, 0). Let Kij = (1/i!j!)(∂ i+j /∂si ∂uj )Kf (s, u)|(s=0,u=0) for 0 ≤ i ≤ 7 and 0 ≤ j ≤ 3. If L0 = 1, L1 = 1, L2 = 2, K00 = 0, K10 = 0, K20 = 0, K30 = 0, K40 = 0, K50 = 0, K60 = 0, K01 = 0, K02 = 0, K03 = 0, K11 = 1, K21 = 2, K12 = 1, K22 = 4, K31 = 1 + L3 , K41 = −4 + 2L3 + L4 are satisfied, then iterative scheme (2) defines a family of eighth-order methods satisfying the error equation below: for n = 0, 1, 2, . . . , en+1 = c2 [−c2 c3 c4 + c33 (K13 − 1) − c23 c4 (L3 − 5) + c22 c32 φ1 + c24 c3 φ2 + c26 φ3 ]e8n + O(e9n ),

(3)

where φ1 = 24 − K32 + 3K13 (L3 − 5) − 2L3 , φ2 = K51 − 2K32 (L3 − 5) + 3K13 (L3 − 5)2 − (L3 − 33)L3 − 2(70 + L4 ) − L5 and φ3 = −5K51 − K70 − K32 (L3 − 5)2 + K13 (L3 − 5)3 + L3 (K51 + 9L3 − 2L4 − L5 ) + 5(45 − 18L3 + 2L4 + L5 ). Proof: The Taylor series expansion of f (xn ) about α up to eighth-order terms with f (α) = 0 leads us to the following: f (xn ) = f (α){en + c2 e2n + c3 e3n + c4 e4n + c5 e5n + c6 e6n + c7 e7n + c8 e8n + O(e9n )}.

(4)

It follows that f (xn ) = f (α){1 + 2c2 en + 3c3 e2n + 4c4 e3n + 5c5 e4n + 6c6 e5n + 7c7 e6n + 8c8 e7n + O(e8n )}.

(5)

For simplicity, we will denote en by e from now on. Symbolic computation of Mathematica [43] yields: f (xn ) = α + c2 e2 − 2(c22 − c3 ) e3 + Y4 e4 + Y5 e5 + Y6 e6n + Y7 e7n + Y8 e8n + O(e9 ), f (xn ) (6) where Y4 = 4c23 − 7c2 c3 + 3c4 , Y5 = −2(4c24 − 10c22 c3 + 3c32 + 5c2 c4 − 2c5 ), Y6 = 16c25 − 52c23 c3 + 33c2 c32 + 28c22 c4 − 17c3 c4 − 13c2 c5 + 5c6 , Y7 = −2[16c26 − 64c24 c3 − 9c33 + 36c23 c4 + 6c42 + 9c22 (7 c32 − 2c5 ) + 11c3 c5 + c2 (−46c3 c4 + 8c6 ) − 3c7 ] and Y8 = 64c27 − 304c25 c3 + 176c24 c4 + 75c32 c4 + c23 (408c32 − 92c5 ) − 31c4 c5 − 27c3 c6 + c22 (−348c3 c4 + 44c6 ) + c2 (−135c33 + 64c42 + 118c3 c5 − 19 c7 ) + 7c8 . yn = xn −

4

M. S. RHEE ET AL.

In view of the fact that f (yn ) = f (xn )|en →(yn −α) , we obtain 8 f (yn ) = f (α)[c2 e2 − 2(c22 − c3 ) e3 + D4 e4 + i=5 Di ei + O(e9 )],

(7)

where D4 = (5c23 − 7c2 c3 + 3c4 ), Di = Di (c2 , c3 , . . . , c8 ) for 5 ≤ i ≤ 8. Hence, we have s=

f (yn ) 7 Ei ei + O(e8 ), = c2 e + (−3c22 + 2c3 ) e2 − 4(8c23 − 10c2 c3 + 3c4 ) e3 + i=4 f (xn )

(8)

where Ei = Ei (c2 , c3 , . . . , c8 ) for 4 ≤ i ≤ 7. Noting that s = O(e) and f (xn )/f (xn ) = O(e), we need a Taylor expansion of Lf (s) about 0 up to seventh-order terms: Lf (s) = L0 + L1 s + L2 s2 + L3 s3 + L4 s4 + L5 s5 + L6 s6 + L7 s7 + O(e8 ),

(9)


where Lj = (dj /dsj )Lf (s) for 0 ≤ j ≤ 7. Thus, we find zn = xn − Lf (s) ·

f (xn ) = α + (1 − L0 )e + c2 (1 − L1 ) e2 f (xn )

8 + [c22 (−2L0 + 4L1 − L2 ) + 2c3 (L0 − L1 )] e3 + i=4 Wi ei + O(e9 ),

where Wi = Wi (c2 , c3 , . . . , c8 , L0 , . . . , L7 ) for 4 ≤ i ≤ 8. By taking L0 = 1, L1 = 1, L2 = 2,

(10)

8 Wi ei + O(e9 ). zn = α − c2 [c22 (L3 − 5) + c3 ] e4 + i=5

(11)

we further obtain In view of the fact that f (zn ) = f (xn )|en →(zn −α) , we obtain 8 Fi ei + O(e9 )], f (zn ) = f (α)[−c2 [c22 (L3 − 5) + c3 ] e4 + i=5

(12)

where Fi = Fi (c2 , c3 , . . . , c8 , L3 , . . . , L7 ) for 5 ≤ i ≤ 8. Hence, we have u=

f (zn ) = [−c3 − c22 (L3 − 5)] e2 + [−2c4 − 4c2 c3 (L3 − 5) + c23 (8L3 − L4 − 26)] e3 f (yn ) 8 + i=4 Gi ei + O(e9 ),

(13)

where Gi = Gi (c2 , c3 , . . . , c8 , L3 , . . . , L7 ) for 4 ≤ i ≤ 8. Noting that f (xn ) = O(e), s = O(e), u = O(e2 ), f (yn ) = O(e2 ) and f (zn ) = O(e4 ), the Taylor expansion of Kf (s, u) about (0, 0) up to seventh-order terms in s and third-order terms in u yields after retaining up to seventh-order terms with K71 = 0, K72 = 0, K73 = 0, K61 = 0, K62 = 0, K63 = 0, K52 = 0, K53 = 0, K43 = 0, K42 = 0, K33 = 0, K23 = 0: Kf (s, u) = K00 + K01 u + K02 u2 + K03 u3 + s(K10 + K11 u + K12 u2 + K13 u3 ) + s2 (K20 + K21 u + K22 u2 ) + s3 (K30 + K31 u + K32 u2 ) + s4 (K40 + K41 u) + s5 (K50 + K51 u) + K60 s6 + K70 s7 + O(e8 ).

(14)

By direct substitution of zn , f (xn ), f (yn ), f (zn ), f (xn ) and Kf (s, u) in Equation (2), we find xn+1 = xn − [Lf (s) + Kf (s, u)] ·

f (xn ) = α − K00 e + c2 (K00 − K10 ) e2 f (xn )

+ [c3 (2K00 + K01 − 2K10 ) − c22 (2K00 + 5K01 − 4K10 + K20 − K01 L3 )] e3 8 + i=4 i ei + O(e9 ),

where i = i (c2 , c3 , . . . , c8 , L3 , . . . , L7 , Kj ), for 4 ≤ i ≤ 8, 0 ≤ j ≤ 7 and 0 ≤ ≤ 3.

(15)


5

By taking K00 = K10 = K01 = K20 = 0

(16)

from Equation (15) along with 4 = 0, we immediately obtain −1 + K11 = 0,

−5 + K30 − K11 (L3 − 5) + L3 = 0,

from which we obtain K11 = 1,

K30 = 0.

(17)


Continuing in this manner at the ith stage with 4 ≤ i ≤ 7, i = 0 and solve i = 0 for remaining Kj to find: K02 = 0,

K21 = 2,

K40 = 0,

K12 = 1,

K03 = 0,

K22 = 4,

K41 = −4 + 2L3 + L4 ,

K31 = 1 + L3 ,

K50 = 0,

K60 = 0.

(18)

By substituting these values of Kj into 8 , we eventually find 8 = c2 [−c2 c3 c4 + c33 (K13 − 1) − c23 c4 (L3 − 5) + c22 c32 φ1 + c24 c3 φ2 + c26 φ3 ],

(19)

with φ1 , φ2 and φ3 as described in Equation (3). This completes the proof.

3. Special cases of weight functions As a result of Theorem 2.1, we easily find Lf (s) and Kf (s, u) in the form of Taylor polynomials as follows: Lf (s) = 1 + 1s + 2s2 + L3 s3 + L4 s4 + L5 s5 + L6 s6 + L7 s7 + O(e8 ), Kf (s, u) = su[1 + u + K13 u2 + 2s(1 + 2u) + s2 (1 + L3 + K32 u) + s3 (−4 + 2L3 + L4 ) + K51 s4 ] + K70 s7 + O(e8 ), (20) where L3 , L4 , L5 , L6 , L7 , K13 , K32 , K51 and K70 may be free parameters. Although various forms of weight functions Lf (s) and Kf (s, u) are applicable, either weight function Lf or Kf is of polynomial type has empirically shown poor convergence as seen in the existing studies by [9,19]. Taking into account the fact that s = O(e), u = O(e2 ) and f (xn )/f (xn ) = O(e), we shall establish eighth-order convergence by restricting ourselves to considering Lf (s) as a family of second-order univariate rational functions and Kf (s, u) as a family of fifth-order bivariate rational functions with real coefficients in the form below: a0 + a1 s + a2 s2 , 1 + b1 s + b2 s2 4 5 i i 2 2 3 3 i=0 ci s + u( i=0 Ai s ) + u (A5 + A6 s + A7 s + A8 s ) + u (A9 + A10 s) Kf (s, u) = , 5 1 + i=1 di si + u( 4i=0 Bi si ) + u2 (B5 + B6 s + B7 s2 + B8 s3 ) + u3 (B9 + B10 s) Lf (s) =

where Ai , Bi , ai , bi , ci , di ∈ R are to be determined for optimal eighth-order convergence.

(21)

6

M. S. RHEE ET AL.

By Theorem 2.1, we let Equation (21) satisfy the constraints (10) and (16)–(18) which give us the coefficients: c0 = 0, c1 = 0, c2 = 0, c3 = 0, c4 = 0, c5 = 0, A0 = 0, A1 = 1, A2 = 2 + d1 , A3 = 5 − 2a2 + b2 + 2d1 + d2 , A4 = 12 + 2a22 + b22 + 5d1 + b2 (6 + d1 ) − a2 (3b2 + 2(6 + d1 )) + 2d2 + d3 , A5 = 0, A9 = 0, B0 = −1 + A6 , B1 = −2 − 2A6 + A7 − d1 . (22) As a result, the reduced form of the desired weight functions is found to be:


Lf (s) = Kf (s, u) =

1 + (a2 − b2 − 1)s + a2 s2 , 1 + (a2 − b2 − 2)s + b2 s2 su[1 + (2 + d1 )s + A3 s2 + A4 s3 + u(A6 + A7 s + A8 s2 ) + A10 u2 ] , 5 1 + i=1 di si + u( 4i=0 Bi si ) + u2 (B5 + B6 s + B7 s2 + B8 s3 ) + u3 (B9 + B10 s)

(23)

where A6 , A7 , A8 , A10 , Bi (2 ≤ i ≤ 10), a2 , b2 , di (1 ≤ i ≤ 5) ∈ R are free parameters. We first observe that weight function Kf (s, u) reduces to Equation (1) studied by Sharma-Arora [37], if given a choice of parameters listed below: A10 = B10 = B9 = B4 = B5 = a2 = b2 = d3 = d4 = d5 = 0, A6 = −1,

A7 = 2,

A8 = −1,

B2 = −2,

B3 = −4,

B6 = 2,

B7 = −7,

B8 = 6,

(24) d1 = −4,

d2 = 4.

For simplified analysis along with a close inspection of Equation (22), we preferably select d4 = d5 = A10 = B10 = 0 and will finally deal with a shortened form of Kf (s, u) as follows: Lf (s) =

1 + (a2 − b2 − 1)s + a2 s2 , 1 + (a2 − b2 − 2)s + b2 s2

su[1 + (2 + d1 )s + A3 s2 + A4 s3 + u(A6 + A7 s + A8 s2 )] , Kf (s, u) = 1 + d1 s + d2 s2 + d3 s3 + u( 4i=0 Bi si ) + u2 (B5 + B6 s + B7 s2 + B8 s3 ) + B9 u3

(25)

where A6 , A7 , A8 , Bi (2 ≤ i ≤ 9), a2 , b2 , d1 , d2 , d3 ∈ R are free parameters with A3 , A4 , B0 , B1 given by Equation (22). Since s = O(e) and u = O(e2 ), we find Kf (s, u) = O(e7 ) from Equation (25), according to which the last sub-step iterative scheme of Equation (2) should give rise to an optimal convergence order of eight with a suitable choice of parameters. For easier analysis, we further take a2 = b2 = B9 = 0 leading to simplified rational weight functions with first-order Lf (s) and fifth-order Kf (s, u) below: Lf (s) = Kf (s, u) =

1−s , 1 − 2s su[1 + (2 + d1 )s + A3 s2 + A4 s3 + u(A6 + A7 s + A8 s2 )] , 1 + d1 s + d2 s2 + d3 s3 + u( 4i=0 Bi si ) + u2 (B5 + B6 s + B7 s2 + B8 s3 )

(26)

where A6 , A7 , A8 , B2 , B3 , B4 , B5 , B6 , B7 , B8 , d1 , d2 , d3 ∈ R are 13 free parameters. Although numerous cases of weight functions satisfying Theorem 2.1 can be constructed, we are especially interested in special cases for which all of the extraneous fixed points (to be discussed in


7

Section 4) of the proposed scheme (2) are purely imaginary. From Equation (35) of Section 4, we desire the governing equation of the extraneous fixed points to take the form of


H(z) =

1 G(t) · , 2(1 + t) (t)

t = z2 ,

(27)

where G(t) = t γ1 (1 + t)γ2 (1 + 3t)γ3 · g(t) and (t) = t σ1 (1 + t)σ2 (1 + 3t)σ3 · w(t) for γ1 , γ2 , γ3 , σ1 , σ2 , σ3 ∈ N. In addition, g(t) and w(t) are polynomials of degree at most 3 and 4, respectively, with γ1 + γ2 + γ3 = 6, and σ1 + σ2 + σ3 = 4. Observe that G(t) and (t) have common factors, which further simplifies the resulting expressions of H(z). The remaining task is again for us to determine appropriate parameters of weight functions in such a way that all the roots of H(z) should be located on the imaginary axis of the complex plane. In Section 4, we shall give an extensive investigation with an appropriate selection of free parameters leading us to purely imaginary extraneous fixed points. To this end, we will seek feasible relationships among the free parameters by imposing some constraints on simplifying the numerator of the resulting expression G(t) to be described in Equation (36). The following cases are of our main interest whose values of (γ1 , γ2 , γ3 ), (σ1 , σ2 , σ3 ) and 10 parameters A6 , A7 , A8 , B2 , B3 , B4 , B5 , B6 , d1 , d2 for each case are discussed in Section 4. Case AX: (γ1 , γ2 , γ3 ) = (1, 2, 3), (σ1 , σ2 , σ3 ) = (1, 1, 2), λ = 4 − A8 + 2d3 , 13 + d3 − 3λ , A7 = −13 − 2d3 + 3λ, A8 = 4 − λ + 2d3 , 2 116 − B7 − B8 + 24d3 − 25λ B8 λ B2 = , B3 = − − 4d3 − , 4 4 2 −4 + B7 + B8 + λ 8 − 4B7 − 3B8 − 2λ , B6 = , B4 = 0, B5 = 4 4 −17 − d3 + 5λ . d1 = 1 − λ, d2 = 2

A6 =

Case AY: (γ1 , γ2 , γ3 ) = (1, 2, 3), (σ1 , σ2 , σ3 ) = (1, 2, 1), λ = 56 − 12B2 − 3B7 − 3B8 + 72d3 , −5 + 2d3 5 − 4d3 λ λ + , A7 = − , A8 = 2d3 , 4 8 2 4 2 − B7 − B8 + 24d3 −50 − B8 − 16d3 λ 3λ B2 = + , B3 = + , B4 = 0, 4 4 4 4 B7 3B8 13 λ 41 − 2d3 B8 5λ + , B6 = −B7 − , d1 = − + , d2 = − . B5 = 4 4 4 2 4 4 8

A6 =

Case AZ: (γ1 , γ2 , γ3 ) = (1, 2, 3), (σ1 , σ2 , σ3 ) = (2, 1, 1), λ = 3(16 − 9A8 + 18d3 ), 181 + 27d3 −181 − 54d3 16 + 18d3 17λ 17λ λ − , A7 = + , A8 = − , 54 324 27 162 9 27 1672 − 27B7 − 27B8 + 648d3 73λ − , B2 = 108 324 B8 + 16d3 −16 + 9B7 + 9B8 λ , B4 = 0, B5 = + , B3 = − 4 36 108 32 − 36B7 − 27B8 23 −209 − 27d3 λ 5λ 25λ B6 = − , d1 = − − , d2 = + . 36 54 27 162 54 324

A6 =

8

M. S. RHEE ET AL.

Case BX: (γ1 , γ2 , γ3 ) = (2, 3, 1), (σ1 , σ2 , σ3 ) = (1, 1, 2), λ = −A8 + 2d3 , d3 λ − , A7 = −1 − 2d3 + 2λ, A8 = −λ + 2d3 , 2 2 18 − B7 − B8 + 24d3 −B8 − 16d3 19λ λ − , B3 = − , B4 = 0, B2 = 4 4 4 2 2 + B7 + B8 −4 − 4B7 − 3B8 −2 − d3 λ λ 5λ − , B6 = + , d1 = −2 − λ, d2 = + . B5 = 4 4 4 2 2 2

A6 =

Case BY: (γ1 , γ2 , γ3 ) = (2, 3, 1), (σ1 , σ2 , σ3 ) = (1, 2, 1), λ = −8 − 4B2 − B7 − B8 + 24d3 , −12 + 12d3 − λ λ , A7 = 2 − 2d3 + , A8 = 2(−1 + d3 ), 24 12 −8 − B7 − B8 + 24d3 − λ B8 λ , B3 = − − 4d3 − , B2 = 4 4 12 B7 + B8 3B8 λ 36 − 12d3 + 5λ , B6 = −B7 − , d1 = −3 − , d2 = . B4 = 0, B5 = 4 4 12 24


A6 =

Case BZ: (γ1 , γ2 , γ3 ) = (2, 3, 1), (σ1 , σ2 , σ3 ) = (2, 1, 1), λ = 2 − 7A8 + 14d3 , 2 + 14d3 − λ 10 3λ 2 + 14d3 − λ , A7 = − − 2d3 + , A8 = , 28 7 14 7 152 − 7B7 − 7B8 + 168d3 − 13λ B8 , B3 = − − 4d3 , B2 = 28 4 16 + 7B7 + 7B8 − λ 8 3B8 λ , B6 = − − B7 − + , B4 = 0, B5 = 28 7 4 14 −26 − λ −38 − 14d3 + 5λ , d2 = . d1 = 14 28

A6 =

Case CX: (γ1 , γ2 , γ3 ) = (1, 3, 2), (σ1 , σ2 , σ3 ) = (1, 1, 2), λ = −8 − 7A8 + 14d3 , −8 + 7d3 − λ 8 − 14d3 + λ −8 + 14d3 − λ , A7 = , A8 = , 14 7 7 −16 − 7B7 − 7B8 + 168d3 − 9λ B8 λ , B3 = − − 4d3 + , B2 = 28 4 2 8 + 7B7 + 7B8 + λ −21B8 − 2(8 + λ) , B6 = −B7 + , B4 = 0, B5 = 28 28 −20 + λ 16 − 7d3 − 5λ , d2 = . d1 = 7 14

A6 =

Case CY: (γ1 , γ2 , γ3 ) = (1, 3, 2), (σ1 , σ2 , σ3 ) = (1, 2, 1), λ = 56 − 4B2 − B7 − B8 + 24d3 , d3 λ λ − , A7 = −4 − 2d3 + , A8 = 2d3 , 2 24 12 56 − B7 − B8 + 24d3 − λ B8 λ , B3 = − − 4d3 − , B2 = 4 4 12 B7 + B8 3B8 λ , B6 = −B7 − , d1 = − , B4 = 0, B5 = 4 4 12

A6 = 2 +

d2 = −6 −

d3 5λ + . 2 24


Case CZ: (γ1 , γ2 , γ3 ) = (1, 3, 2), (σ1 , σ2 , σ3 ) = (2, 1, 1), λ = −40 − 49A8 + 98d3 , 32 + 98d3 − 9λ −32 + 9λ −40 + 98d3 − λ , A7 = −2d3 + , A8 = , 196 98 49 704 − 49B7 − 49B8 + 1176d3 − 51λ B8 , B3 = − − 4d3 , B2 = 196 4 40 + 49B7 + 49B8 + λ 20 3B8 λ , B6 = − − B7 − − , B4 = 0, B5 = 196 49 4 98 5(40 + λ) −176 − 98d3 + 25λ d1 = − , d2 = . 98 196

A6 =


Case DX: (γ1 , γ2 , γ3 ) = (3, 2, 1), (σ1 , σ2 , σ3 ) = (1, 1, 2), λ = −A8 + 2d3 , d3 − 9λ , A7 = −1 − 2d3 + 6λ, A8 = 2d3 − λ, 2 18 − B7 − B8 + 24d3 − 43λ B8 λ , B3 = − − 4d3 − , B4 = 0, B2 = 4 4 2 2 + B7 + B8 + 7λ 3B8 7λ , B6 = −1 − B7 − − , d1 = −2 − λ, B5 = 4 4 2

A6 =

d2 =

−2 − d3 + 5λ . 2

Case DY: (γ1 , γ2 , γ3 ) = (3, 2, 1), (σ1 , σ2 , σ3 ) = (1, 2, 1), λ = A8 − 2(1 + d3 ), 60 + 12d3 + 17λ 23λ , A7 = −6 − 2d3 − , A8 = 2 + 2d3 + λ, 24 12 56 − B7 − B8 + 24d3 + 15λ B8 λ B2 = , B3 = − − 4d3 − , 4 4 12 B7 + B8 3B8 5λ −84 − 12d3 − 25λ , B6 = −B7 − , d1 = −1 + , d2 = . B4 = 0, B5 = 4 4 12 24

A6 =

Case DZ: (γ1 , γ2 , γ3 ) = (3, 2, 1), (σ1 , σ2 , σ3 ) = (2, 1, 1), λ = 2 − A8 + 2d3 , 10 + 2d3 − 5λ 5λ , A7 = −6 − 2d3 + , A8 = 2 + 2d3 − λ, 4 2 56 − B7 − B8 + 24d3 − 19λ B8 , B3 = − − 4d3 , B4 = 0, B2 = 4 4 B7 + B8 + λ −4B7 − 3B8 − 2λ λ −14 − 2d3 + 5λ , B6 = , d1 = −1 − , d2 = . B5 = 4 4 2 4

A6 =

Case EX: (γ1 , γ2 , γ3 ) = (2, 2, 2), (σ1 , σ2 , σ3 ) = (1, 1, 2), λ = −A8 + 2d3 , 2 + d3 − 3λ , A7 = −2 − 2d3 + 3λ, A8 = 2d3 − λ, 2 24 − B7 − B8 + 24d3 − 25λ B8 λ , B3 = − − 4d3 − , B4 = 0, B2 = 4 4 2 B7 + B8 + λ −4B7 − 3B8 − 2λ −2 − d3 + 5λ , B6 = , d1 = −2 − λ, d2 = . B5 = 4 4 2

A6 =

9

10

M. S. RHEE ET AL.

Case EY: (γ1 , γ2 , γ3 ) = (2, 2, 2), (σ1 , σ2 , σ3 ) = (1, 2, 1), λ = 24 − 4B2 − B7 − B8 + 24d3 , d3 λ λ − , A7 = −2 − 2d3 + , A8 = 2d3 , 2 24 12 24 − B7 − B8 + 24d3 − λ B8 λ , B3 = − − 4d3 − , B4 = 0, B2 = 4 4 12 B7 + B8 3B8 λ d3 5λ , B6 = −B7 − , d1 = −2 − , d2 = −1 − + . B5 = 4 4 12 2 24

A6 = 1 +

Case EZ: (γ1 , γ2 , γ3 ) = (2, 2, 2), (σ1 , σ2 , σ3 ) = (2, 1, 1), λ = 8 − A8 + 2d3 , 44 + 2d3 − 5λ 5λ , A7 = −22 − 2d3 + , A8 = 8 + 2d3 − λ, 4 2 176 − B7 − B8 + 24d3 − 19λ B8 , B3 = − − 4d3 , B2 = 4 4 −8 + B7 + B8 + λ 3B8 λ B4 = 0, B5 = , B6 = 4 − B7 − − , 4 4 2 λ −44 − 2d3 + 5λ . d1 = 2 − , d2 = 2 4


A6 =

Case FX: (γ1 , γ2 , γ3 ) = (1, 4, 1), (σ1 , σ2 , σ3 ) = (1, 1, 2), λ = 64 + 25A8 − 50d3 , −64 + 25d3 + λ 19 2λ −64 + 50d3 + λ , A7 = − 2d3 − , A8 = , 50 5 35 25 343 B7 93λ B8 − − + 6d3 + , B2 = − 50 4 4 700 B8 λ 14 + 175B7 + 175B8 − λ , B3 = − − 4d3 − , B4 = 0, B5 = 4 14 700 1 3B8 −686 − λ 266 − 35d3 + λ λ + , d1 = , d2 = . B6 = − − B7 − 25 4 350 175 70

A6 =

Case FY: (γ1 , γ2 , γ3 ) = (1, 4, 1), (σ1 , σ2 , σ3 ) = (1, 2, 1), λ = −18 − 7A8 + 14d3 , −68 + 28d3 − 3λ 26 5λ 18 λ , A7 = − 2d3 + , A8 = − + 2d3 − , 56 7 28 7 7 −184 − 7B7 − 7B8 + 168d3 − 11λ , B2 = 28 −B8 − 16d3 + λ B7 + B8 3B8 , B4 = 0, B5 = , B6 = −B7 − , B3 = 4 4 4 −108 + λ 204 − 28d3 − 5λ , d2 = . d1 = 28 56

A6 =

Case FZ: (γ1 , γ2 , γ3 ) = (1, 4, 1), (σ1 , σ2 , σ3 ) = (2, 1, 1), λ = −338 − 119A8 + 238d3 , 158 + 238d3 + 23λ 404 − 476d3 − 15λ −338 + 238d3 − λ , A7 = , A8 = , 476 238 119 26 101λ B7 B8 − − + 6d3 + , B2 = 119 4 4 476 B8 −32 + 17B7 + 17B8 − λ 64 − 68B7 − 51B8 + 2λ , B6 = , B3 = − − 4d3 , B4 = 0, B5 = 4 68 68 −566 + 11λ −26 − 238d3 − 55λ d1 = , d2 = . 238 476

A6 =


11

4. Extraneous fixed points and their dynamics We in this section will devote ourselves to investigating the extraneous fixed points [42] of iterative map (2) and relevant dynamics associated with their basins of attraction. The dynamics underlying basins of attraction was initiated by Stewart [39] and followed by works of Amat et al., e.g. [2,3], Andreu et al. [4], Argyros-Magreñan [5], Chun et al. [12], Chicharro et al. [8], Chun-Neta [10], Cordero et al. [15], Geum et al. [19,20], Magreñan [27,28], Neta et al. [30–32] and Scott et al. [35]. We usually locate a zero α of a nonlinear equation f (x) = 0 by means of a fixed point ξ of iterative methods of the form


xn+1 = Rf (xn ),

n = 0, 1, . . . ,

(28)

where Rf is the iteration function under consideration. In general, Rf might possess other fixed points ξ = α. Such fixed points are called the extraneous fixed points of the iteration function Rf . It is well known that extraneous fixed points may result in attractive, indifferent or repulsive cycles as well as other periodic orbits influencing the dynamics underlying the basins of attraction. Exploration of such dynamics as well as discovery of its complicated behaviour gives us a valuable motivation of the current analysis. In connection with proposed family of methods (2), we obtain a more specific form of iterative map (28) as follows: xn+1 = Rf (xn ) = xn −

f (xn ) Hf (xn ), f (xn )

(29)

where Hf (xn ) = Lf (s) + Kf (s, u) can be regarded as a weight function of the classical Newton’s method. It is obvious that α is a fixed point of Rf . The points ξ = α for which Hf (ξ ) = 0 are extraneous fixed points of Rf . For ease of analysis of the relevant dynamics, we restrict ourselves to considering only combinations of weight functions Lf (s) and Kf (s, u) in the form of univariate and bivariate rational functions as described by Equation (21). A special attention will be paid to some selected cases to be shown later in this section in order to pursue further properties of their extraneous fixed points and relevant dynamics associated with their basins of attraction. The existence of such extraneous fixed points would affect the global iteration dynamics, which was demonstrated for simple zeros via König functions and Schröder functions [42] applied to a family of functions {fk (x) = xk − 1, k ≥ 2} according to the joint work of Vrscay and Gilbert [42] published in 1988. Especially, the presence of attractive cycles induced by the extraneous fixed points of Rf may alter the basins of attraction due to the trapped sequence {xn }. Even in the case of repulsive or indifferent fixed points, an initial value x0 chosen near a desired root may converge to another unwanted remote root. Indeed, these aspects of the Schröder functions were observed in an application to the same family of functions {fk (x) = xk − 1, k ≥ 2}. For simplified dynamics related to the extraneous fixed points underlying the basins of attraction for iterative maps (29), we first choose a simple quadratic polynomial from the family of functions {fk (x) = xk − 1, k ≥ 2}. By closely following the works of Chun et al. [9,13] and Neta et al. [29,30,32], we then construct Hf (xn ) = Lf (s) + Kf (s, u) in Equation (29). We now apply a prototype quadratic polynomial f (z) = (z2 − 1) to Hf (xn ) and construct H(z), with a change of a variable t = z2 , in the form of N (t) H(z) = , (30) D(t) where both D(t) and N (t) are polynomial functions of t with no common factors. Since H is a rational function, it would be preferable for us to deal with the underlying dynamics of iterative map (29) on the Riemann sphere [6] where points ‘0 (zero)’ and ‘∞’ can be treated as the desired extraneous fixed points. If such points arise, we are interested in only the finite extraneous fixed point 0 under which the relevant dynamics can be described in a region containing the origin by investigating the attractor basins associated with iterative map (29).

12

M. S. RHEE ET AL.

Indeed, the extraneous fixed points ξ of Rf in Equation (29) can be found from the roots t of H(z) with z = t1/2 via relation below: t 1/2 if t = 0, ξ= (31) 0(double root) if t = 0.


4.1. Purely imaginary extraneous fixed points We now pay a special attention to the dynamics underlying purely imaginary extraneous fixed points of iterative map (29). One should be aware that the boundary of two basins of attraction of two roots for the prototype quadratic polynomial f (z) = z2 − 1 is the imaginary axis of the complex plane. Hence, it is worth to explore how the extraneous fixed points on the imaginary axis influence the dynamical behaviour of iterative map (29). It is our important task to find a possible combination of Lf and Kf leading to purely imaginary extraneous fixed points, whose investigation was done by Chun et al. [13]. As a preliminary task, we first describe the following lemma regarding the negative real roots of a quadratic equation, which would play a role in determining the desired purely imaginary extraneous fixed points in connection with the prototype quadratic polynomial f (z) = z2 − 1. As a preliminary task, we first describe the following lemma regarding the negative real roots of a cubic equation for later use in characterizing the cubic g(t) described by Equation (39). Lemma 4.1: Let q(x) = ax3 + bx2 + cx + d be a cubic equation with real coefficients a = 0, b,c,d satisfying D ≥ 0, where D = 18abcd + b2 c2 − 4b3 d − a(4c3 + 27ad2 ). Let r1 , r2 and r3 be the three roots of q(x) = 0. Then all three roots r1 < 0, r2 < 0 and r3 < 0 hold if and only if all four coefficients a,b,c,d have the same sign. Proof: In view of the elementary theory of a cubic equation [38,41], the hypothesis D ≥ 0 guarantees that all the roots of q(x) = 0 are real. Suppose that r1 < 0, r2 < 0 and r3 < 0. Then via Vieta’s formula we find −b/a = r1 + r2 + r3 < 0 and c/a = r1 r2 + r2 r3 + r3 r1 > 0 and −d/a = r1 r2 r3 < 0. We easily obtain ab > 0, ac > 0 and ad > 0. Hence, all four coefficients a,b,c,d have the same sign. Conversely, we first suppose that all four coefficients a, b, c, d have the same sign, yielding ab > 0, ac > 0 and ad > 0. Then Vieta’s formula again yields three relations −b/a = r1 + r2 + r3 < 0, c/a = r1 r2 + r2 r3 + r3 r1 > 0 and −d/a = r1 r2 r3 < 0. Substituting r3 = (1/r1 r2 )(−d/a) from the last relation, we have the two remaining relations below: 1 d r1 + r2 + − < 0, r1 r2 a (32) r1 + r2 d − > 0. r1 r 2 + r1 r2 a If r1 + r2 > 0 held true, then the first relation of Equation (32) multiplied by the negative real number ‘−(r1 + r2 )’ would give −(r1 + r2 )2 − ((r1 + r2 )/r1 r2 )(−d/a) > 0. Adding this to the second relation of Equation (32), we obtain −(r1 + r2 )2 + r1 r2 > 0. Adding (r1 + r2 )2 ≥ 4r1 r2 , we find r1 r2 ≤ 0 giving r1 + r2 + (1/r1 r2 )(−d/a) > 0, which contradicts the first relation of Equation (32). Hence, r1 + r2 ≤ 0 must hold. If r1 + r2 = 0 held true, then it would give r1 r2 = −r12 > 0 contradictory to the fact that −r12 ≤ 0. Therefore, we must have r1 + r2 < 0, which yields r1 r2 > 0 from the second relation of Equation (32). Consequently, r1 < 0 and r2 < 0. Furthermore, r3 = (1/r1 r2 )(−d/a) < 0. This completes the proof. Remark 4.1: The proof of the converse of the above theorem can be made alternatively by the use of Descartes’ Rule of Signs [41]. The number of sign variations in the sequence of coefficients of q(x) is found to be exactly zero. By virtue of Descartes’ Rule of Signs, q(x) has no positive real roots, i.e. all


13

its roots are non-positive. Since d has the same sign as a = 0, it gives nonzero roots of q(x), implying that all the roots are negative. To begin the detailed study regarding the purely imaginary extraneous fixed points, we now employ weight function Lf in (26) applied to f (z) = (z2 − 1): 1 1 1− 2 , s= 4 z 2 1 3z + 1 Lf = . 2 z2 + 1

(33)

Besides, we are able to express Kf in terms of z and free parameters A6 , A7 , A8 , B2 , B3 , B4 , B5 , B6 , B7 , B8 , d1 , d2 , d3 with the use of


u=

1 (z2 − 1)2 . · 4 (z2 + 1)2

(34)

Although such lengthy expression of Kf is not explicitly shown here, the simplified second-order form of Lf will greatly reduce the complexity of Kf as well as the desired Hf = Lf + Kf given by Equation (30). As a result, the explicit form of the relevant H(z) given by Equation (30) becomes H(z) =

G(t; β0 , β1 , . . . , β9 ) 1 · , 2(1 + t) (t; ω0 , ω1 , . . . , ω8 )

(35)

where G(t; β0 , β1 , . . . , β9 ) and (t; ω0 , ω1 , . . . , ω8 ) are concisely denoted by G(t) and (t), respectively, as below: G(t) =

9

βi t i ,

(36)

i=0

with β0 = 24 + B4 + 10d1 + 4d2 + 2d3 , β1 = −112 − 2A8 − 4B3 − B4 − B8 − 46d1 − 20d2 − 22d3 , β2 = 4(36 + 2A7 + 3A8 + 4B2 − 2B4 + B7 + B8 + 16d1 + 20d2 − 16d3 ), β3 = 4(52 + 24 A6 − 26A7 − 7A8 + 4B2 + 8B3 + 4B4 − 4B6 − 3B7 − 28d1 + 92d2 + 4d3 )t 3 , β4 = 2(224 + 320 A6 − 28A7 + 14A8 − 56B2 − 16B3 + B4 + 32B5 + 16B6 − 6B7 − 14B8 − 830d1 + 124d2 + 90d3 ), β5 = −2(−3096 + 16A6 − 140A7 − 8B2 + 20B3 + 13B4 + 32B5 − 40B6 − 50B7 − 35B8 + 1550 d1 + 236d2 − 42d3 ), β6 = −4(−4764 + 256A6 − 54A7 + 7A8 − 44B2 − 16B3 − 4B4 + 96B5 + 80B6 + 45B7 + 21B8 + 252d1 + 188d2 + 44d3 ), β7 = −4(−6076 + 120A6 + 94A7 − 7A8 + 20 B2 − 2B4 − 224B5 − 100B6 − 39B7 − 14B8 − 656d1 + 20d2 + 32d3 ), β8 = 13, 096 + 384A6 − 168 A7 − 12A8 − 80B2 − 32B3 − 11B4 − 704B5 − 224B6 − 68B7 − 20B8 + 2594d1 + 420d2 + 58d3 , β9 = 2176 + 416A6 + 200A7 + 2A8 + 48B2 + 12B3 + 3B4 + 192B5 + 48B6 + 12B7 + 3B8 + 634 d1 + 204d2 + 50d3 , and (t) =

8

ωi t i ,

(37)

i=0

with ω0 = B4 , ω1 = −4B3 − 4B4 − B8 − 16d3 , ω2 = 16B2 + 12B3 + 4B4 + 4B7 + 7B8 + 64d2 − 16d3 , ω3 = 128 + 128A6 − 64A7 − 32B2 − 4B3 + 4B4 − 16B6 − 24B7 − 21B8 − 192d1 + 128d2 + 48d3 , ω4 = 640 + 128A6 + 64A7 − 16B2 − 20B3 − 10B4 + 64B5 + 80B6 + 60B7 + 35B8 − 832 d1 − 64d2 + 48d3 , ω5 = 3840 − 256A6 + 128A7 + 64B2 + 20B3 + 4B4 − 256B5 − 160B6 − 80 B7 − 35B8 − 640d1 − 256d2 − 48d3 , ω6 = 6912 − 256A6 − 128A7 − 16B2 + 4B3 + 4B4 + 384 B5 + 160B6 + 60B7 + 21B8 + 640d1 − 64d2 − 48d3 , ω7 = 4224 + 128A6 − 64A7 − 32B2 − 12 B3 − 4B4 − 256B5 − 80B6 − 24B7 − 7B8 + 832d1 + 128d2 + 16d3 , ω8 = 640 + 128A6 + 64A7 + 16B2 + 4B3 + B4 + 64B5 + 16B6 + 4B7 + B8 + 192d1 + 64d2 + 16d3 .

14

M. S. RHEE ET AL.

Note that the weight function Lf (z) = 12 ((1 + 3t)/(1 + t)) with t = z2 contains two factors (1 + 3t) and (1 + t). Hence we naturally consider a special case of H(z) in the form of a simplified rational function possibly with such two factors. To this end, we construct Hf = Lf + Kf =

1 G(t) , 2(1 + t) (t)

(38)

where G(t) and (t) may involve some of such factors in addition to a factor t corresponding to the origin (considered as purely imaginary) of the complex plane, as shown below: G(t) = tγ1 (1 + t)γ2 (1 + 3t)γ3 · g(t) σ1

σ2

σ3

(t) = t (1 + t) (1 + 3t)

· w(t)

for γ1 , γ2 , γ3 ∈ N, for σ1 , σ2 , σ3 ∈ N,

γ1 + γ2 + γ3 = 6, σ1 + σ2 + σ3 = 4,

(39)


where g(t) and w(t) are polynomials of degree at most 3 and 4, respectively. The expression of H(z) in Equation (35) will be further simplified as follows: H(z) =

1 γ1 −σ1 g(t) (1 + t)γ2 −σ2 −1 (1 + 3t)γ3 −σ3 · ·t 2 w(t)

with t = z2 .

(40)

If we further restrict with γ2 ≥ 2, then all possible combinations of (γ1 , γ2 , γ3 ) are listed by {(1, 2, 3), (2, 3, 1), (1, 3, 2), (3, 2, 1), (2, 2, 2), (1, 4, 1)}. Since all possible combinations of (σ1 , σ2 , σ3 ) are listed by {(1, 1, 2), (1, 2, 1), (2, 1, 1)}, we are able to construct 18 different combinations of G(t) and (t). For systematic numbering of all possible 18 cases, we assign not only six letters A, B, C, D, E, F to the six triplets of (γ1 , γ2 , γ3 ) listed by {(1, 2, 3), (2, 3, 1), (1, 3, 2), (3, 2, 1), (2, 2, 2), (1, 4, 1)} in order, but also three letters X, Y, Z to the three triplets of (σ1 , σ2 , σ3 ) listed by {(1, 1, 2), (1, 2, 1), (2, 1, 1)} in order. Hence, combining two letters covers all possible 18 cases. Consequently, Cases AX, AY, . . . , FZ shall denote the respective cases when (γ1 , γ2 , γ3 , σ1 , σ2 , σ3 ) = (1, 2, 3, 1, 1, 2), (γ1 , γ2 , γ3 , σ1 , σ2 , σ3 ) = (1, 2, 3, 1, 2, 1), . . . , (γ1 , γ2 , γ3 , σ1 , σ2 , σ3 ) = (1, 4, 1, 2, 1, 1). In order to obtain purely imaginary extraneous fixed points, we further require that all the roots of g(t) should be negative. Let g(t) = q0 + q1 t + q2 t 2 + q3 t 3 and w(t) = p0 + p1 t + p2 t 2 + p3 t 3 + p4 t 4 . Then, the roots of g(t) = 0 would contribute to the desired extraneous fixed points. In view of the fact that γ1 + γ2 + γ3 = 6 and σ1 + σ2 + σ3 = 4, the forms of Equation (39) would require a set of six constraints 0 = G(0) = G (0) = · · · G(γ1 −1) (0) = G(−1) = G (−1) = · · · G(γ2 −1) (−1) = G(− 13 ) = G (− 13 ) = · · · G(γ3 −1) (− 13 )

(41)

and additionally a set of four constraints 0 = (0) = (0) = · · · (σ1 −1) (0) = (−1) = (−1) = · · · (σ2 −1) (−1) = (− 13 ) = (− 13 ) = · · · (σ3 −1) (− 13 ).

(42)

Since G(−1) = −256(8B5 + 4B6 + 2B7 + B8 ), (−1) = 128(8B5 + 4B6 + 2B7 + B8 ), we find that G(−1) = −2 (−1), from which G(−1) = 0 implies (−1) = 0. Consequently, the above 10 constraints reduce to 9 constraints. For the four Cases AY, BY, CY, EY, we can solve these nine constraints for nine parameters A6 , A7 , A8 , B3 , B4 , B5 , B6 , d1 , d2 in terms of at most 4 remaining parameters B2 , B7 , B8 , d3 . For remaining 12 cases, we can solve the corresponding 9 constraints for 9 parameters A6 , A7 , B2 , B3 , B4 , B5 , B6 , d1 , d2 in terms of at most 4 remaining parameters A8 , B7 , B8 , d3 . Due to the fact that σ1 ≥ 1, we immediately find that B4 = 0 from the first equation (0) = 0 = B4 of


15

Equation (42). If we substitute these nine parameters back into G(t) and (t) in Equation (39), the explicit forms of g(t) and w(t) with their coefficients in terms of at most four remaining parameters A8 (or B2 ), B7 , B8 , d3 for a given combination of (γ1 , γ2 , γ3 ) and (σ1 , σ2 , σ3 ). If a new parameter λ is conveniently introduced as an appropriate affine combination of d3 , A8 (or B2 ), B7 , B8 , then all 18 Cases AX, AY, . . . , FZ, 10 parameters A6 , A7 , A8 , B2 , B3 , B4 , B5 , B6 , d1 , d2 can be expressed in terms of four parameters d3 , B7 , B8 , λ. After a tedious algebra, the resulting parameters for all 18 cases are already described at the end of Section 3. The following proposition is useful in the analysis of proposed family of methods (2) in both computational and dynamics aspects.


Proposition 4.2: For each case, all coefficients of g(t) and w(t) can be expressed as an affine combination of λ. Proof: Since one proof is similar to another, it suffices to consider a typical case AX with (γ1 , γ2 , γ3 ) = (1, 2, 3), (σ1 , σ2 , σ3 ) = (1, 1, 2) and λ = 4 − A8 + 2d3 . Solving the 9 constraints for (γ1 , γ2 , γ3 ) = (1, 2, 3) and (σ1 , σ2 , σ3 ) = (1, 1, 2), we find that A6 = 12 (1 + 3A8 − 5d3 ), A7 = −1 − 3A8 + 4d3 , B2 = 14 (16 + 25A8 − B7 − B8 − 26d3 ), B3 = −2 + A8 /2 − B8 /4 − 5d3 , B4 = 0, B5 = 1 1 1 4 (−A8 + B7 + B8 + 2d3 ), B6 = 4 (2A8 − 4B7 − 3B8 − 4d3 ), d1 = −3 + A8 − 2d3 , d2 = 2 (3 − 5A8 + 9d3 ). Substituting such nine coefficients into G(t) and (t), we find g(t) = 4[(7 + 4A8 − 8d3 )t 3 + (35 + 8A8 − 16d3 )t 2 − 3(−7 + 4A8 − 8d3 )t + 1] w(t) = 2[(12 + 7A8 − 14d3 )t 4 + t 3 (88 + 28A8 − 56d3 ) − 14(A8 − 2(4 + d3 ))t 2 − 20(A8 − 2(1 + d3 ))t + 4 − A8 + 2d3 ]. Applying A8 = 4 − λ + 2d3 to the above equations yields: g(t) = −4[t 3 (4λ − 23) + t2 (8λ − 67) − 3t(4λ − 9) − 1] w(t) = 2[t4 (40 − 7λ) − 4t3 (7λ − 50) + 14t2 (4 + λ) + 20t(λ − 2) + λ],

completing the proof.

Remark 4.3: If we express w(t) at t = 0 as a scalar multiple of λ, i.e. w(0) = hλ, h ∈ R, then each of the remaining cases shows that all coefficients of g(t) and w(t) can be expressed as an affine combination of λ. Each selection of λ is shown at the end of Section 3. Special λ-values for interesting forms of H(z) are listed in Table 1. We are further interested in possible extraneous fixed points from the roots of the cubic equation denoted by g(t) = q0 + q1 t + q2 t 2 + q3 t 3

(43)

with qi = qi (λ), (0 ≤ i ≤ 3). The discriminant D of g(t) can be expressed in terms of parameter λ. We denote a set D = {λ ∈ R : D ≥ 0}.

(44)

B = {λ ∈ R : q3 q2 > 0 and q3 q1 > 0 and q3 q0 > 0}

(45)

We further denote a set

16

M. S. RHEE ET AL.

Table 1. H(z) and λ for the selected values of γ = (γ1 , γ2 , γ3 ) and σ = (σ1 , σ2 , σ3 ). Case AX1

γ

σ

g(t) w(t)

λ

H(z), t = z2

(1, 1, 2)

2[−1 − 3t(−9 + 4λ) + t2 (−67 + 8λ) + t3 (−23 + 4λ)] , λ + 20t(−2 + λ) + 14t2 (4 + λ) + ν3 t3 + ν4 t4

13 3

ν3 = 4(50 − 7λ), ν4 = 40 − 7λ

4

(1 + 3t)(3 + 75t + 97t2 + 17t3 ) 13 + 140t + 350t2 + 236t3 + 29t4 (1 + 21t + 35t2 + 7t3 ) 4(1 + t)(1 + 6t + t2 ) (1 + 3t)(1 + 33t + 27t2 + 3t3 ) (5 + 10t + t2 )(1 + 10t + 5t2 ) 4(1 + 3t)2 (1 + 17t + 25t2 + 5t3 ) (1 + t)(15 + 156t + 578t2 + 684t3 + 103t4 )

AX2 AX6 AY4

5 (1, 2, 3) (1, 2, 1)

AY5

2[λ − 2 + t(86 + 5λ) − 5t (−50 + λ) + t (50 − λ)] , 2 λ + 4t(16 + λ) + t2 (448 + 13λ) + ν3 t3 + ν4 t4 3 1 ν3 = 12(48 − λ), ν4 = (256 − 5λ) 3 2

3

AY6


BX1

12

16 20

(1, 1, 2)

BX6 BY6

(2, 3, 1) (1, 2, 1)

BZ5

(2, 1, 1)

CX4

(1, 1, 2)

CY1

(1, 3, 2) (1, 2, 1)

32(1 + 6t + t2 )(t(−4 + λ) − λ) (1 + 33t + 27t2 + 3t3 )(t(−4 + λ) − λ) 32(1 + 6t + t2 ) = 1 + 33t + 27t2 + 3t3 32[t3 (28 − 3λ) + 7t2 (20 − λ) + 7t(12 + λ) + 3λ + 4] , ν4 t4 + ν3 t3 + 14t2 (32 + λ) + 4t(16 + 5λ) + λ ν4 = 64 − 7λ, ν3 = 28(16 − λ) 4[t3 (228 + λ) + t2 (1396 + 13λ) − t(13λ − 172) − λ − 4] , 4 1 4 t (256 + λ) + t3 (608 + 5λ) + ν2 t2 + ν1 t − λ 7 7 ν1 = 4(32 − λ), ν2 = 2(256 + λ) 4[t3 (57 + λ) + t2 (349 + 13λ) − t(13λ − 43) − λ − 1] −λ + 4t(8 − λ) + 2t2 (64 + λ) + ν3 t3 + ν4 t4 4 1 ν3 = (152 + 5λ), ν4 = (64 + λ) 7 7 −8[t3 (69 − λ) + t2 (201 − 2λ) + 3t(−27 + λ) + 3] , λ + 20t(−24 + λ) + 14t2 (48 + λ) + ν3 t3 + ν4 t4 ν3 = 4(600 − 7λ), ν4 = 480 − 7λ

CY2 CY6

0 2 24

16

1 + 21t + 35t2 + 7t3 4(1 + t)(1 + 6t + t2 )

−8

1 + 21t + 35t2 + 7t3 ) 4(1 + t)(1 + 6t + t2 )

56 40 60

DX2

(1, 1, 2)

DY7

(3, 2, 1) (1, 2, 1)

64[t (2 − 3λ) + t (14 − 3λ) + 7t(2 + λ) − λ + 2] ν4 t4 + ν3 t3 + 2t2 (66 + 37λ) + t(4 − 16λ) + λ ν4 = (12 − 19λ), ν3 = (108 − 40λ) 3

2

−

2 7

16[3t3 (28 + 3λ) + t2 (420 + 37λ) − 21t(λ − 12) + 12 − 25λ] λ − 28tλ + t2 (384 − 226λ) + ν3 t3 + ν4 t4

−4

DY9

ν3 = 4(576 + 53λ), ν4 = 384 + 41λ

0

EX2

8[t (7 − 3λ) − 7t (−5 + λ) + 7t(3 + λ) + 3λ + 1] ν4 t4 + ν3 t3 + 14t2 (8 + λ) + 4t(4 + 5λ) + λ

2 3

ν4 = 16 − 7λ, ν3 = 28(4 − λ)

0

(1, 1, 2)

3

EX6 EY6

(2, 2, 2) (1, 2, 1)

EZ1

(2, 1, 1)

8[t (42 − λ) − 3t (−70 + λ) + 3(2 + λ) + t(126 + λ)] , λ + 4t(24 + 5λ) + 2t2 (432 + 7λ) + ν3 t3 + ν4 t4 ν3 = 4(456 − 7λ), ν4 = 288 − 7λ 3

EZ7 FY4 FY5 FY6

(1, 4, 1) (1, 2, 1)

2

2

2

4[t3 (23 − 2λ) + t2 (67 − 4λ) + 3t(−9 + 2λ) + 1] , λ + 20t(−4 + λ) + 14t2 (8 + λ) + ν3 t3 + ν4 t4

44 5

ν3 = 400 − 28λ, ν4 = 80 − 7λ

10

4[t (228 + λ) + t (1396 + 13λ) − t(13λ − 172) − λ − 4] , −λ + 4t(32 − λ) + 2t2 (256 + λ) + ν3 t3 + ν4 t4 4 1 ν3 = (608 + 5λ), ν4 = (256 + λ) 7 7 3

2

8t(1 + 3t)2 (2 + 5t + t2 ) (1 + t)(1 + 36t + 158t2 + 276t3 + 41t4 ) 8t(1 + 3t)2 (2 + 5t + t2 ) (1 + t)(1 + 36t + 158t2 + 276t3 + 41t4 ) 16t(1 + t)(1 + 6t + t2 ) (1 + 3t)(1 + 33t + 27t2 + 3t3 ) 16t(1 + t)(1 + 6t + t2 ) (1 + 3t)(1 + 33t + 27t2 + 3t3 ) 2t(7 + 35t + 21t2 + t3 ) 1 + 28t + 70t2 + 28t3 + t4

−46 −32 −18

(1 + 3t)(3 + 87t + 89t2 + 13t3 ) 2(7 + 80t + 182t2 + 104t3 + 11t4 ) (1 + 3t)(3 + 39t + 121t2 + 29t3 ) 2(5 + 40t + 154t2 + 160t3 + 25t4 ) (1 + 3t)(1 + 33t + 27t2 + 3t3 ) (5 + 10t + t2 )(1 + 10t + 5t2 ) 64t2 (4 + 21t + 26t2 + 5t3 ) (1 + t)(1 + 3t)(−1 + 31t + 357t2 + 61t3 ) 32t2 (7 + 14t + 3t2 ) −1 + 28t + 322t2 + 364t3 + 55t4 1 + 21t + 35t2 + 7t3 4(1 + t)(1 + 6t + t2 ) 2t(9 + 77t + 91t2 + 15t3 ) 1 + 44t + 182t2 + 140t3 + 17t4 1 + 21t + 35t2 + 7t3 4(1 + t)(1 + 6t + t2 ) 8t(1 + 3t)(3 + 5t)(1 + 9t + 2t2 ) (1 + t)(1 + 68t + 446t2 + 884t3 + 137t4 ) (1 + 3t)(5 + 129t + 159t2 + 27t3 ) 2(11 + 120t + 294t2 + 192t3 + 23t4 ) (1 + 3t)(1 + 33t + 27t2 + 3t3 ) (5 + 10t + t2 )(1 + 10t + 5t2 ) 2(1 + t)(3 + t)(1 + 18t + 13t2 ) 23 + 156t + 210t2 + 108t3 + 15t4 1 + 21t + 35t2 + 7t3 4(1 + t)(1 + 6t + t2 ) 2(1 + t)(1 + 3t)(1 + 26t + 5t2 ) 9 + 100t + 238t2 + 148t3 + 17t4


17

whose elements make all four coefficients q0 , q1 , q2 , q3 have the same sign. We now use Lemma 4.1 to locate all three negative roots of g(t) = 0 for purely imaginary extraneous fixed points. After a lengthy algebraic process, we are able to find the desired sets D, B and D ∩B containing λ-values for which purely imaginary extraneous fixed points can be located. One should note that extraneous fixed point zeros ξ = 0 (being considered as purely imaginary) may be found on the boundary of B. Let B¯ denote the closure of B. According to interesting val¯ we classify the sub-cases of each case from Cases AX, AY, . . . , FZ by appending ues of λ ∈ D ∩ B, sequential arabic numerals such as Cases AX1, AX2, . . . , FX2, . . . . Presented below are values of (γ1 , γ2 , γ3 ), (σ1 , σ2 , σ3 ), λ, g(t), D, B and D ∩ B for each case under consideration. Case AX: (γ1 , γ2 , γ3 ) = (1, 2, 3), (σ1 , σ2 , σ3 ) = (1, 1, 2), λ = 4 − A8 + 2d3 .


(1) g(t) = 4 [t 3 (−23 + 4λ) + t2 (−67 + 8λ) − 3t(−9 + 4λ) − 1]. (2) D = {λ : λ ≤ 0.938575 or λ ≥ 3.29184}, B = {λ : 2.25 < λ < 5.75}. (3) D ∩ B = {λ : 3.29184 ≤ λ < 5.75}. 17 116 7 23 The seven sub-cases AX1–AX7 are identified with λ ∈ { 13 3 , 4, 5 , 25 , 2 , 5, 4 } in order. Case AY: (γ1 , γ2 , γ3 ) = (1, 2, 3), (σ1 , σ2 , σ3 ) = (1, 2, 1), λ = 56 − 12B2 − 3B7 − 3B8 + 72d3 .

(1) g(t) = 2 [t 3 (50 − λ) − 5t2 (−50 + λ) + t(86 + 5λ) + λ − 2]. (2) D = {λ : λ ≤ 20.8093 or λ ≥ 50}, B = {λ : 2 < λ < 50}. (3) D ∩ B = {λ : 2 < λ ≤ 20.8093}. The six sub-cases AY1–AY6 are identified with λ ∈ {2, 4, 8, 12, 16, 20} in order. Case AZ: (γ1 , γ2 , γ3 ) = (1, 2, 3), (σ1 , σ2 , σ3 ) = (2, 1, 1), λ = 3(16 − 9A8 + 18d3 ). 4 (1) g(t) = 81 [t 3 (1203 − 11λ) + t 2 (4287 − 19λ) + t(−327 + 31λ) + 21 − λ]. (2) D = {λ : λ ≤ −16.6790 or λ ≥ 17.7879}, B = {λ : 10.5484 < λ < 21}. (3) D ∩ B = {λ : 17.7879 ≤ λ < 21}.

The two sub-cases AZ1, AZ2 are identified with λ ∈ {18, 21} in order. Case BX: (γ1 , γ2 , γ3 ) = (2, 3, 1), (σ1 , σ2 , σ3 ) = (1, 1, 2), λ = −A8 + 2d3 . (1) g(t) = −64 (1 + 6t + t 2 )(t(λ − 4) − λ). (2) D = R, B = {λ : 0 < λ < 4}. (3) D ∩ B = {λ : 0 < λ < 4}. 2 The eight sub-cases BX1–BX8 are identified with λ ∈ {0, 12 , 36 38 , 5 , 1, 2, 3, 4} in order. Case BY: (γ1 , γ2 , γ3 ) = (2, 3, 1), (σ1 , σ2 , σ3 ) = (1, 2, 1), λ = −8 − 4B2 − B7 − B8 + 24d3 .

(1) g(t) = − 83 [3t 3 (−28 + λ) + 7t 2 (−60 + λ) − 7t(36 + λ) − 3(4 + λ)]. (2) D = R, B = {λ : −4 < λ < 28}. (3) D ∩ B = {λ : −4 < λ < 28}. The seven sub-cases BY1–BY7 are identified with λ ∈ {−4, 0, 4, 6, 12, 24, 28} in order. Case BZ: (γ1 , γ2 , γ3 ) = (2, 3, 1), (σ1 , σ2 , σ3 ) = (2, 1, 1), λ = 2 − 7A8 + 14d3 . 3 2 (1) g(t) = 16 7 [t (114 − λ) + t (698 − 13λ) + t(86 + 13λ) + λ − 2]. (2) D = {λ : λ ≤ 24.7402 or λ ≥ 83.0709}, B = {λ : 2 < λ < 698 13 }. (3) D ∩ B = {λ : 2 ≤ λ ≤ 24.7402}.

18

M. S. RHEE ET AL.

38 152 The six sub-cases BZ1–BZ6 are identified with λ ∈ {2, 20 3 , 5 , 13 , 16, 24} in order. Case CX: (γ1 , γ2 , γ3 ) = (1, 3, 2), (σ1 , σ2 , σ3 ) = (1, 1, 2), λ = −8 − 7A8 + 14d3 .

(1) g(t) = − 87 [t 3 (−57 − λ) + t 2 (−349 − 13λ) + t(−43 + 13λ]) + λ + 1]. (2) D = {λ : λ ≤ −41.5354 or λ ≥ −12.3701}, B = {λ : −26.8462 < λ < −1}. (3) D ∩ B = {λ : −12.3701 ≤ λ < −1}. 23 9 The seven sub-cases CX1–CX7 are identified with λ ∈ {− 16 9 , −12, − 2 , −8, − 2 , − 72 , −1} in order. Case CY: (γ1 , γ2 , γ3 ) = (1, 3, 2), (σ1 , σ2 , σ3 ) = (1, 2, 1), λ = 56 − 4B2 − B7 − B8 + 24d3 .


(1) g(t) = 83 [t 3 (69 − λ) + t 2 (201 − 2λ) + 3t(−27 + λ) + 3]. (2) D = {λ : λ ≤ 11.2629 or λ ≥ 39.5021}, B = {λ : 27 < λ < 69}. (3) D ∩ B = {λ : 39.5021 ≤ λ < 69}. The seven sub-cases CY1–CY7 are identified with λ ∈ {56, 40, 42, 48, 54, 60, 69} in order. Case CZ: (γ1 , γ2 , γ3 ) = (1, 3, 2), (σ1 , σ2 , σ3 ) = (2, 1, 1), λ = −40 − 49A8 + 98d3 . 8 (1) g(t) = 49 [t 3 (607 − 13λ) + t 2 (2683 − 15λ) + t(−163 + 29λ) − λ + 9]. (2) D = {λ : λ ≤ −11.0311 or λ ≥ 8.37459}, B = {λ : 5.62069 < λ < 9}. (3) D ∩ B = {λ : 8.37459 ≤ λ < 9}.

The two sub-cases CZ1, CZ2 are identified with λ ∈ { 17 2 , 9} in order. Case DX: (γ1 , γ2 , γ3 ) = (3, 2, 1), (σ1 , σ2 , σ3 ) = (1, 1, 2), λ = −8 − 7A8 + 14d3 . (1) g(t) = 128 [t 3 (2 − 3λ) + t 2 (14 − 3λ) + 7t(2 + λ) − λ + 2]. (2) D = {λ : λ ≤ −8.82797 or λ ≥ −0.367756}, B = {λ : −2 < λ < 23 }. (3) D ∩ B = {λ : −0.367756 ≤ λ < 23 }. 1 2 The seven sub-cases DX1–DX7 are identified with λ ∈ { 16 , − 27 , 25 , 18 43 , − 3 , 0, 3 } in order. Case DY: (γ1 , γ2 , γ3 ) = (3, 2, 1), (σ1 , σ2 , σ3 ) = (1, 2, 1), λ = A8 − 2(1 + d3 ). 3 2 (1) g(t) = 16 3 [3t (28 + 3λ) + t (420 + 37λ) − 21t(−12 + λ) + 12 − 25λ]. (2) D = {λ : λ ≤ −8.64561 or λ ≥ −6}, B = {λ : − 28 3 < λ < 0.48}. < λ ≤ −8.64561 or − 6 ≤ λ < 0.4869}. (3) D ∩ B = {λ : − 28 3 56 84 28 The nine sub-cases DY1–DY9 are identified with λ ∈ {− 24 17 , − 15 , − 25 , − 3 , −9, −6, −4, −2, 0} in order. Case DZ: (γ1 , γ2 , γ3 ) = (3, 2, 1), (σ1 , σ2 , σ3 ) = (2, 1, 1), λ = 2 − A8 + 2d3 .

(1) g(t) = 32 [t 3 (14 − 3λ) − 7t2 (−10 + λ) + 7t(6 + λ]) + 3λ + 29]. (2) D = R, B = {λ : − 23 < λ < 14 3 }. }. (3) D ∩ B = {λ : − 23 < λ < 14 3 The nine sub-cases DZ1–DZ9 are identified with in order.

56 14 2 14 λ ∈ { 12 5 , 19 , 5 , − 3 , 0, 1, 2, 4, 3 }


19

Case EX: (γ1 , γ2 , γ3 ) = (2, 2, 2), (σ1 , σ2 , σ3 ) = (1, 1, 2), λ = −A8 + 2d3 . (1) g(t) = 16 [t 3 (7 − 3λ) − 7t2 (−5 + λ) + 7t(3 + λ) + 3λ + 1]. (2) D = R, B = {λ : − 13 < λ < 73 }. (3) D ∩ B = {λ : − 13 < λ < 73 }. 2 1 7 The eight sub-cases EX1–EX8 are identified with λ ∈ { 16 , 23 , 24 25 , 5 , − 3 , 0, 1, 3 } in order. Case EY: (γ1 , γ2 , γ3 ) = (2, 2, 2), (σ1 , σ2 , σ3 ) = (1, 2, 1), λ = 24 − 4B2 − B7 − B8 + 24d3 .

(1) g(t) = 83 [t 3 (42 − λ) − 3t2 (−70 + λ) + t(126 + λ) + 3(2 + λ)]. (2) D = {λ : λ ≤ −8.64561 or λ ≥ −6}, B = {λ : −2 < λ < 42}. (3) D ∩ B = {λ : −2 < λ ≤ 6.18678}.


1 24 The six sub-cases EY1–EY6 are identified with λ ∈ { 24 , 5 , −2, 0, 2, 6} in order. Case EZ: (γ1 , γ2 , γ3 ) = (2, 2, 2), (σ1 , σ2 , σ3 ) = (2, 1, 1), λ = 8 − A8 + 2d3 .

(1) g(t) = 16 [t 3 (23 − 2λ) + t2 (67 − 4λ) + 3t(−9 + 2λ) + 1]. (2) D = λ ≤ 1.87715 or λ ≥ 6.58369, B = {λ : 92 < λ < 23 2 }. }. (3) D ∩ B = {λ : 6.58369 ≤ λ < 23 2 176 23 The eight sub-cases EZ1–EZ8 are identified with λ ∈ { 44 5 , 19 , 4, 7, 8, 9, 10, 2 } in order. Case FX: (γ1 , γ2 , γ3 ) = (1, 4, 1), (σ1 , σ2 , σ3 ) = (1, 1, 2), λ = 64 + 25A8 − 50d3 . 32 (1) g(t) = 175 [t 3 (658 + 3λ) + t2 (4382 − 13λ) + t(574 + 9λ) − 14 + λ]. (2) D = λ ≤ 132.648 or λ ≥ 2565.89, B = {λ : 14 < λ < 337.0769}. (3) D ∩ B = {λ : 14 < λ ≤ 132.6478}. 4802 The six sub-cases FX1–FX6 are identified with λ ∈ {64, 133 2 , 93 , 14, 60, 132} in order. Case FY: (γ1 , γ2 , γ3 ) = (1, 4, 1), (σ1 , σ2 , σ3 ) = (1, 2, 1), λ = −18 − 7A8 + 14d3 .

(1) g(t) = 47 [t 3 (−228 − λ) + t 2 (−1396 − 13λ) + t(−172 + 13λ) + λ + 4]. (2) D = {λ : λ ≤ −166.1417 or λ ≥ −49.4805}, B = {λ : −107.3846 < λ < −4}. (3) D ∩ B = {λ : −49.4805 ≤ λ < −4}. 104 184 The nine sub-cases FY1–FY9 are identified with λ ∈ {− 68 3 , − 5 , − 11 , −46, −32, −18, −14, −10, −4} in order. Case FZ: (γ1 , γ2 , γ3 ) = (1, 4, 1), (σ1 , σ2 , σ3 ) = (2, 1, 1), λ = −338 − 119A8 + 238d3 . 8 (1) g(t) = 119 [t 3 (−5426 − 109λ) + t2 (−10, 618 + 39λ) + t(874 + 73λ) − 3λ − 62]. (2) D = λ ≤ −18.8202 or λ ≥ 18.8087, B = {λ : −20.6667 < λ < −11.9726}. (3) D ∩ B = {λ : −20.6667 < λ ≤ −18.8202}.

The three sub-cases FZ1–FZ3 are identified with λ ∈ {− 62 3 , −20, −19} in order. Despite the availability of rich sub-cases considered thus far, we typically list (γ1 , γ2 , γ3 ), (σ1 , σ2 , σ3 ), g(t)/w(t), λ and H(z) in Table 1, for selected 25 sub-cases AX1, AX2, AX6, AY4, AY5, AY6, BX1, BX6, BY6, BZ5, CX4, CY1, CY2, CY6, DX2, DY7, DY9, EX2, EX6, EY6, EZ1, EZ7, FY4, FY5, FY6 with simplified forms of Kf (s, u). Besides, the extraneous fixed points for the selected 25 sub-cases are listed in Table 2 together with those extraneous fixed points of existing method SA. In view of the analysis done so far and a close inspection of Table 1, the following remark is useful.

20

M. S. RHEE ET AL.

Remark 4.4: (i) Once λ is chosen, we have freedom to select parameters d3 , B7 , B8 . Note that H(z) can be obtained without specifying parameter values of d3 , B7 , B8 for all selected cases. (ii) Three cases (AX6, CY6, EZ7) (highlighted in yellow) give the same H(z) = (1 + 3t)(1 + 33t + 27t2 + 3t3 )/{(5 + 10t + t2 )(1 + 10t + 5t2 )} which is also the same as SA, two cases (BX1, BX6) (highlighted in green) the same H(z) = 16t(1 + t)(1 + 6t + t2 )/{(1 + 3t)(1 + 33t + 27t 2 + 3t3 )}, and six cases (AX2, BZ5, CX4, DY9, EX6, FY5) (highlighted in cyan) the same H(z) = (1 + 21t + 35t2 + 7t3 )/{4(1 + t)(1 + 6t + t2 )}.


4.2. Stability of extraneous fixed points As a result of the case studies pursued thus far for f (z) = z2 − 1, we include in Table 2 the desired purely imaginary extraneous fixed points in typical sub-cases. By direct computation of absolute values of multipliers Rf (ξ ) for iterative map (29) with f (z) = z2 − 1, we find that all of the purely imaginary extraneous fixed points ξ of H in each of the listed cases in Table 2 are found to be indifferent except for extraneous fixed point double 0. The extraneous fixed point double 0 for each of Cases BX1, BX6, BY6, EX2, EY6 is found to be repulsive and highlighted by a framed-value. Interestingly, no case with attractive extraneous fixed points has been found. The following proposition describes the details of stabilities of the multipliers for the all the cases AX, AY, . . . , FZ. Proposition 4.5: Let ±ξ be the extraneous fixed points obtained from the expression √ g(t)/w(t) of H(z) in Equation(40). Then stabilities of the possible extraneous fixed points 0, ±i, ±i/ 3 and ±ξ for the 18 cases AX, AY, . . . , FZ are characterized by the following: Table 2. Extraneous fixed points ξ and their stability for selected cases. Case

ξ

AX1 AX2 AX6 AY4 AY5 AY6 BX1

±2.18932i, ±0.932983i, ±i/ 3, ±0.205661i ±2.07652i, ±0.797473i, √±0.228243i ±2.74748i, ±1.19175i, √ ±i/ 3, ±0.176327i ±2.01802i, ±0.922879i, ±i/√ 3(double), ±0.275446i ±1.92767i, ±1.09135i, ±i/ √3(double), ±0.305019i ±1.73205i, ±1.37638i, ±i/ 3(double)i, ±0.32492i ±2.41421i, ±i, ±0.414214i, 0 (double)

BX6

No. of ξ

√

±2.41421i, ±i, ±0.414214i,

0 (double)

8 6 8 10 10 10 8 8

BY6 BZ5 CX4 CY1 CY2 CY6 DX2 DY7 DY9

±4.38129i, ±1.25396i, ±0.481575i, 0 (double) ±2.07652i, ±0.797473i, ±0.228243i ±2.07652i, ±0.797473i, √±0.228243i ±2.38198i, ±1.06609i, ±i/ 3, ±0.189172i √ ±0.348006i ±1.95657i, ±i/ 3, ±0.472367i, √ ±2.74748i, ±1.19175i, ±i/ 3, ±0.176327i ±2.06341i, ±0.809824i, ±0.535264i, 0(quadruple) ±2.02415i, ±0.754652i, 0(quadruple) ±2.07652i, ±0.797473i, ±0.228243i

8 6 6 8 8 8 10 8 6

EX2 EX6

±2.25373i, ±0.920924i, ±0.373208i, 0 (double) ±2.07652i, ±0.797473i, ±0.228243i √ ±2.13578i, ±0.662153i, ±i/ 3(double), 0 (double) √ ±2.21963i, ±0.959862i, ±i/√ 3, ±0.201983i ±2.74748i, ±1.19175i, ±i/ 3, ±0.176327i ±1.73205i, ±1.15179i, ±i, ±0.240798i ±2.07652i, ±0.797473i, √ ±0.228243i ±2.27184i, ±i, ±i/ 3,√±0.196851i ±2.74748i, ±1.19175i, ±i/ 3, ±0.176327i

8 6

EY6 EZ1 EZ7 FY4 FY5 FY6 SA

10 8 8 8 6 8 8

Note: In this table, most extraneous fixed points are indifferent, while boxed-values are repulsive extraneous fixed points. Interestingly, no attractive extraneous fixed points exist for the selected cases.


21

√ (1) The extraneous fixed points quadruple 0, ±i (simple or double), ±i/ 3 (simple or double) and ±ξ are all found to be indifferent. (2) The extraneous fixed point double 0 is found to be repulsive. Proof: To prove (1) and (2), it should suffice to take several typical cases AX, AY, BX, DY, FX, BY, DZ, EX, EY as follows: √ (i) Case AX for extraneous fixed points ±i/ 3 (simple) and ±ξ . The corresponding H(z) for Case AX found to be: H(z) =

(1 + 3t)[−1 + t(27 − 12λ) + t2 (8λ − 67) + t3 (4λ − 23)] , −λ − 20t(λ − 2) − 14t2 (4 + λ) + 4t3 (7λ − 50) + t4 (7λ − 40)

(46)


where t = z2 and λ is described earlier in Section 4.1. Besides, the derivative of iterative map Rf in Equation (29) is given by (t − 1)[−1 + t(22 − 10λ) − 10t2 (λ + 2) + 2t 3 (9λ − 59) + t4 (2λ − 11)] . (47) 2t[−λ − 20t(λ − 2) − 14t 2 (λ + 4) + 4t 3 (7λ − 50) + t4 (7λ − 40)] √ By direct substitution of the extraneous fixed points z = ±i/ 3 (simple), i.e. t = − 13 into Rf (z), we √ immediately find Rf (±i/ 3) = 1. We now let the extraneous fixed points ±ξ satisfy Rf (z) =

−1 + t(27 − 12λ) + t2 (8λ − 67) + t3 (4λ − 23) = 0 with t = ξ 2 . For brevity, we first denote the left side of the above equation by dλ (t) = −1 + t(27 − 12λ) + t2 (8λ − 67) + t3 (4λ − 23). Then the second factor of the numerator of Equation (47) is given by −1 + t(22 − 10λ) − 10t2 (λ + 2) + 2t 3 (9λ − 59) + t4 (2λ − 11) = qλ (t) · dλ (t) + rλ (t) = rλ (t), where qλ (t) =

1977 − 664λ + 56λ2 + t(253 − 90λ + 8λ2 ) (23 − 4λ)2

and 8(−181 + 60λ − 5λ2 + t(5186 − 4028λ + 910λ2 − 64λ3 )+ t 2 (−14, 381 + 7056λ − 1161λ2 + 64λ3 ) rλ (t) = − . (23 − 4λ)2 Hence, Equation (47) at this extraneous fixed points ±ξ becomes Rf (z) =

(t − 1)rλ (t) . 2t[−λ − 20t(λ − 2) − 14t 2 (λ + 4) + 4t 3 (7λ − 50) + t4 (7λ − 40)]

(48)

Since (t − 1)rλ (t) − 2t[−λ − 20t(λ − 2) − 14t2 (λ + 4) + 4t 3 (7λ − 50) + t4 (7λ − 40)] =−

2dλ (t)[−4(181 − 60λ + 5λ2 ) + t(1920 − 655λ + 56λ2 ) + t 2 (920 − 321λ + 28λ2 )] =0 (−23 + 4λ)2

in view of the fact dλ (t) = 0, we find Rf (±ξ ) = 1. √ (ii) Case AY for extraneous fixed points ±i/ 3 (double).

22

M. S. RHEE ET AL.

The corresponding H(z) and Rf (z) for Case AY are found to be: H(z) =

(1 + 3t)2 [2 − λ + t(−86 − 5λ) + 5t2 (λ − 50) + t 3 (λ − 50)] , −3λ − 12t(16 + λ) + t2 (−896 − 26λ) + 36t 3 (−48 + λ) + t 4 (−256 + 5λ)

(49) (t − 1)[2 − λ − 6t(12 + λ) − 4t2 (109 + 4λ) + t4 (−62 + λ) + 22t 3 (−44 + λ)] = . 2t[t 2 (−896 − 26λ) + 36t 3 (−48 + λ) − 3λ − 12t(16 + λ) + t4 (−256 + 5λ)] √ By direct substitution of the extraneous fixed points z = ±i/ 3 (double), i.e. t = − 13 into Rf (z), we √ immediately find Rf (±i/ 3) = 1. (iii) Case BX for extraneous fixed points ±i and 0 (double). The corresponding H(z) and Rf (z) for Case BX are found to be: Rf (z)


H(z) = Rf (z)

16t(1 + t)(1 + 6t + t2 ) , (1 + 3t)(1 + 33t + 27t 2 + 3t3 )

(50)

(t − 1)(7 + 35t + 21t2 + t 3 ) = . (1 + 3t)(1 + 33t + 27t 2 + 3t3 )

By direct substitution of the extraneous fixed points z = ±i and 0 (double), i.e. t = −1 and t = 0, respectively, into Rf (z), we immediately find Rf (±i) = 1 and Rf (0) = −7, respectively. (iv) Case DY for extraneous fixed point 0 (quadruple). The corresponding H(z) and Rf (z) for Case DY are found to be: H(z) = Rf (z)

8t2 [12 − 25λ − 21t(−12 + λ) + t2 (420 + 37λ) + t 3 (84 + 9λ)] , (1 + t)[λ − 28tλ + t2 (384 − 226λ) + 4t3 (576 + 53λ) + t 4 (384 + 41λ)]

(t − 1)[−λ + t(48 − 73λ) + t2 (672 + 69λ) + t 3 (48 + 5λ)] = . λ − 28tλ + t 2 (384 − 226λ) + 4t3 (576 + 53λ) + t 4 (384 + 41λ)

(51)

By direct substitution of the extraneous fixed point 0 (quadruple), i.e. t = 0(double) into Rf (z), we immediately find Rf (0) = 1. (v) Case FX for extraneous fixed point ±i (double). The corresponding H(z) and Rf (z) for Case FX are found to be: H(z) =

8(1 + t)2 [−14 + t(574 + 9λ) + t 2 (4382 − 13λ) + λ + t3 (658 + 3λ)] , (1 + 3t)[25λ + 4t(2919 + 4λ) + t2 (21, 532 − 38λ) + t 3 (10, 612 − 8λ) + 5t 4 (196 + λ)]

Rf (z) =

(t − 1)[4(λ − 14) + t(2072 + 27λ) + 4t2 (3661 + λ) + t 3 (20, 132 − 38λ) + 7700t 4 + t 5 (308 + 3λ)] . t(1 + 3t)[25λ + 4t(2919 + 4λ) + t 2 (21, 532 − 38λ) + t 3 (10, 612 − 8λ) + 5t 4 (196 + λ)]

(52)

Rf (z),

we By direct substitution of the extraneous fixed point ±i (double), i.e. t = −1(double) into immediately find Rf (±i) = 1. (vi) Cases BY, DZ, EX, EY for extraneous fixed point 0 (double). The corresponding H(z) and Rf (z) for these cases can be similarly found as obtained so far. Here, we list their respective multipliers at double 0 by means of λ as follows: ⎧ −5 − 24/λ for − 4 < λ < 28, ⎪ ⎪ ⎪ ⎨−5 − 4/λ for − 2/3 < λ < 14/3, Rf (0) = (53) ⎪−5 − 2/λ for − 1/3 < λ < 7/3, ⎪ ⎪ ⎩ −5 − 12/λ for − 2 < λ ≤ 6.18678. After a close examination, we find that |Rf (0)| > 1, implying the repulsiveness of these multipliers.


23

The stabilities of remaining cases can be similarly shown as those of the above typical cases, completing the proof. Remark 4.6: Among all selected cases, no case with attractive extraneous fixed points has been found. It is interesting to observe that the extraneous fixed point double 0 is found to be repulsive, while the extraneous fixed point quadruple 0 is found to be indifferent throughout the selected cases. In case that f (z) is a generic polynomial rather than z2 − 1, it would be certainly interesting to investigate the dynamics underlying the relevant extraneous fixed points. However, due to the increased algebraic complexity, we would encounter difficulties in describing the dynamics underlying the extraneous fixed points. An effective way of exploring such dynamics is to illustrate basins of attraction under iterative map (29) with f (z) as a generic polynomial. We will illustrate the basins of attraction to pursue the dynamics of the iterative map Rp of the form


zn+1 = Rp (zn ) = zn −

p(zn ) Hp (zn ), p (zn )

(54)

for a generic polynomial p(zn ) and a weight function Hp (zn ). Indeed, basins of attraction for the fixed points or the extraneous fixed points as well as their attracting periodic orbits would reflect complex dynamics whose illustrative description will be made for various polynomials in the latter part of Section 5. Before closing this section, we prefix the iterative maps in Table 2 corresponding to cases AX1, AX2, AX6, AY4, AY5, AY6, BX1, BX6, BY6, BZ5, CX4, CY1, CY2, CY6, DX2, DY7, DY9, EX2, EX6, EY6, EZ1, EZ7, FY4, FY5, FY6 with W for later use in describing the relevant dynamics. In addition, we identify map SA for method (1).

5. Numerical experiments and complex dynamics In this section, we first deal with computational aspects of proposed family of methods (2) for a variety of test functions along with an existing competitive method SA; then we discuss the dynamics underlying extraneous fixed points based on iterative maps (54) by illustrating the relevant basins of attraction. In Section 4, we were able to find extraneous fixed points using λ-values without specifying parameters d3 , B7 , B8 . For numerical experiments in both computational and dynamical aspects, we need to provide all 10 coefficients A6 , A7 , A8 , B2 , B3 , B4 , B5 , B6 , d1 , d2 of Kf (s, u) for a given λ. For simplified Kf (s, u), we set d3 = B7 = B8 = 0. Table 3 shows the desired parameter values and Kf (s, u) for the selected cases AX1, AX2, AX6, AY4, AY5, AY6, BX1, BX6, BY6, BZ5, CX4, CY1, CY2, CY6, DX2, DY7, DY9, EX2, EX6, EY6, EZ1, EZ7, FY4, FY5, FY6. Each case has been implemented to verify the theoretical convergence. Later on in this section, we will explore the complex dynamics with the use of illustrated basins of attraction of selected rational iterative maps WAX1 through WFY6 and an existing method SA. A number of numerical experiments have been implemented with Mathematica programming to confirm the developed theory. Throughout these experiments, we have maintained 160 digits of minimum number of precision, via Mathematica command $MinPrecision = 160, to achieve the specified accuracy. In case that α is not exact, it is replaced by a more accurate value which has more number of significant digits than the preassigned number $MinPrecision = 160. Definition 5.1 (Computational convergence order): Assume that theoretical asymptotic error constant η = limn→∞ (|en |/|en−1 |p ) and convergence order p ≥ 1 are known. Define pn = log |en /η|/log |en−1 |) as the computational convergence order. Note that limn→∞ pn = p. Remark 5.1: Note that pn requires knowledge at two points xn , xn−1 , while the usual COC (computational order of convergence) log(|xn − xn−1 |/|xn−1 − xn−2 |)/log(|xn−1 − xn−2 |/|xn−2 − xn−3 |)

24

M. S. RHEE ET AL.


Table 3. Parameter values of λ, A6 , A7 , A8 , B2 , B3 , B4 , B5 , B6 , d1 , d2 and Kf (s, u) for all selected cases as well as SA. Case

λ

(A6 , A7 , A8 )

(B2 , B3 , B4 , B5 , B6 )

(d1 , d2 )

Kf (s, u)

AX1

13 3

(0, 0, − 13 )

13 1 1 ( 23 12 , − 6 , 0, 12 , − 6 )

7 (− 10 3 , 3)

4su(3 − 4s − s2 (u − 2)) 12 − 12u + u2 − 2s(20 − 8u + u2 ) + s2 (28 + 23u) − 26s3 u

AX2

4

( 12 , −1, 0)

(4, −2, 0, 0, 0)

(−3, 32 )

su(2 + u − 2s(1 + u) + s2 ) 2 − u − 2s(3 + u) + s2 (3 + 8u) − 4s3 u

AX6

5

(−1, 2, −1)

(− 94 , − 52 , 0, 14 , − 12 )

(−4, 4)

4su(1 − s)2 (1 − u) 4 − 8u + u2 − 2s(8 − 12u + u2 ) − s2 (9u − 16) − 10s3 u

AY4

12

( 13 , − 23 , 0)

( 11 3 , −3, 0, 0, 0)

7 (− 10 3 , 3)

su(3 + u − 2s(2 + u) + 2s2 ) (1 − s)(3 − 2u + 9s2 u − s(7 + 2u))

AY5

16

( 16 , − 13 , 0)

( 10 3 , −4, 0, 0, 0)

(− 11 3 ,

AY6

20

(0, 0, 0)

(3, −5, 0, 0, 0)

(−4, 4)

BX1

0

(0, −1, 0)

( 92 , 0, 0, 12 , −1)

(−2, −1)

BX6

2

(−1, 3, −2)

(−5, −1, 0, 0, 0)

(−4, 4)

BY6

24

(− 32 , 4, −2)

(−8, −2, 0, 0, 0)

(−5,

BZ5

16

(− 12 , 2, −2)

(−2, 0, 0, 0, 0)

(−3, 32 )

CX4

−8

(0, 0, 0)

(2, −4, 0, 0, 0)

(−4, 4)

CY1

56

(− 13 , 23 , 0)

(0, − 14 3 , 0, 0, 0)

(− 14 3 ,

CY2

40

( 13 , − 23 , 0)

(4, − 10 3 , 0, 0, 0)

7 (− 10 3 , 3)

CY6

60

(− 12 , 1, 0)

(−1, −5, 0, 0, 0)

(−5,

DX2

− 27

2 ( 97 , − 19 7 , 7)

1 ( 53 7 , 7 , 0, 0, 0)

12 (− 12 7 ,− 7 )

su(7 + 9u + s(2 − 19u) + s2 (2u − 1)) 7 + 2u − 3s(4 + 13u) + s2 (53u − 12) + s3 u

DY7

−4

(− 13 , 53 , −2)

(−1, 13 , 0, 0, 0)

(− 83 , 23 )

su(3 − u − s(2 − 5u) − s2 (6u − 1)) 3 − 4u + s(9u − 8) + s2 (2 − 3u) + s3 u

DY9

0

( 52 , −6, −2)

(14, 0, 0, 0, 0)

(−1, − 72 )

su(2 + 5u + 2s(1 − 6u) + s2 (4u − 1)) 2 + 3u − 2s(1 + 12u) + 7s2 (4u − 1)

EX2

2 3

(0, 0, − 23 )

1 1 1 ( 11 6 , − 3 , 0, 6 , − 3 )

(− 83 , 23 )

EX6

0

(1, −2, 0)

(6, 0, 0, 0, 0)

(−2, −1)

EY6

6

( 34 , − 32 , 0)

( 92 , − 12 , 0, 0, 0)

(− 52 , 14 )

su(4 + 3u − 2s(1 + 3u) + s2 ) 4 − u − 10s(1 + u) + s2 (1 + 18u) − 2s3 u

EZ1

44 5

(0, 0, − 45 )

1 2 ( 11 5 , 0, 0, 5 , − 5 )

(− 12 5 , 0)

su(5 − 2s − s2 (4u − 1)) 5 − 5u + u2 − 2s(6 − u + u2 ) + 11s2 u

EZ7

10

(− 32 , 3, −2)

(− 72 , 0, 0, 12 , −1)

(−3, 32 )

su(2 − 3u − 2s(1 − 3u) − s2 (4u − 1)) 2 − 5u + u2 − 2s(3 − 7u + u2 ) − s2 (7u − 3)

FY4

−46

( 54 , − 92 , 4)

23 ( 23 2 , − 2 , 0, 0, 0)

(− 11 2 ,

FY5

−32

( 12 , −2, 2)

(6, −8, 0, 0, 0)

(−5,

13 2 )

su(2 + u − 2s(3 + 2u) + s2 (3 + 4u)) 2 − u − 10s + s2 (13 + 12u) − 16s3 u

FY6

−18

(− 14 , 12 , 0)

( 12 , − 92 , 0, 0, 0)

(− 92 ,

21 4 )

su(4 − u + 2s(u − 5) + 5s2 ) 4 − 5u + 2s(7u − 9) + s2 (21 + 2u) − 18s3 u

SA

N/A∗

(−1, 2, −1)

(−2, −4, 2, 0, 0)

19 6 )

13 2 )

17 3 )

13 2 )

31 4 )

(−4, 4)

su(6 + 5s2 + u − 2s(5 + u)) 6 − s(22 − 6u) − 5u + s2 (19 + 20u) − 24s3 u su(1 − s)2 1 + 2s(u − 2) − u + s2 (4 + 3u) − 5s3 u 2su(1 − su) 2 − 2u + u2 − 2s(2 + u + u2 ) + s2 (9u − 2) su(1 − s)(1 − u + s(2u − 1)) 1 − 2u − s(4 − 7u) − s2 (5u − 4) − s3 u su(2 − 3u − 2s(3 − 4u) − s2 (4u − 3)) 2 − 5u − 10s(1 − 2u) − s2 (16u − 13) − 4s3 u su(2 − u − 2s(1 − 2u) − s2 (4u − 1)) 2 − 3u − 2s(3 − 4u) − s2 (4u − 3) su(1 − s)2 (1 − 2s)(1 − u − 2s + 2s2 u) su(2s − 1)(2s + u − 3) 3 − 4u + 2s(6u − 7) + 17s2 − 14s3 u su(3 + u − 2s(2 + u) + 2s2 ) (1 − s)(3 − 2u − s(7 + 2u) + 10s2 u) su(2 − u + 2s(u − 3) + 3s2 ) 2 − 3u + 10s(u − 1) − s2 (2u − 13) − 10s3 u

2su(3 − 2s − s2 (2u − 1)) 6 − 6u + u2 − 2s(8 − 2u + u2 ) + s2 (4 + 11u) − 2s3 u su(1 + u − 2su) 1 − 2s(1 + 2u) + s2 (6u − 1)

su(4 + 5u − 2s(7 + 9u) + s2 (7 + 16u)) 4 + u − 2s(11 + 7u) + s2 (31 + 46u) − 46s3 u

su(1 − u)(1 − s)2 (1 − 2s)(1 − su)(1 − 2s − 2u + 3su)

Note: For all above cases other than SA use d3 = B7 = B8 = 0, while SA uses d3 = 0, B7 = −7, B8 = 6. Three cases AX6, CY6, EZ7 (highlighted in yellow) have different forms of Kf but show identical H(z) as that of SA. ∗ N/A = not available.


25

does require knowledge at four points xn , xn−1 , xn−2 , xn−3 . Hence, pn can be handled with a less number of working precision digits than the usual COC whose number of working precision digits is at least p times as large as that of pn . Computed values of xn are accurate with up to $MinPrecision significant digits. If α has the same accuracy of $MinPrecision as that of xn , then en = xn − α would be nearly zero and hence computing |en+1 |/|en |p would unfavourably break down. To clearly observe the convergence behaviour, we desire α to have more significant digits that are digits higher than $MinPrecision. To supply such α, a set of following Mathematica commands are used: sol = FindRoot[f (x), {x, x0 }, PrecisionGoal → + $MinPrecision, WorkingPrecision → 2 ∗ $MinPrecision];


α = sol[[1, 2]] In this experiment, we assign = 16. As a result, the numbers of significant digits of xn and α are found to be 160 and 176, respectively. Nonetheless, we list both of them with up to 15 significant digits for proper readability. The error bound = 12 × 10−120 is assigned to satisfy |xn − α| < . Typical methods WAX2, WBX1, WDY7 have been successfully implemented with test functions F1 − F3 below:

π WAX2 : F1 (x) = cos + x2 − π, α ≈ 1.81648572902222, x √ √ π 1 11 WBX1 : F2 (x) = x − 3x3 cos + 2 − + 4 3, α = 2, x+1 x +1 5 √ 3 19 WDY7 : F3 (x) = − log (x − 2)2 + , α = 2−i , 16 4 where log z(z ∈ C) represents a principal analytic branch such that − π < Im(log z) ≤ π. Table 4 clearly confirms eighth-order convergence. The values of computational asymptotic error constant agree up to 10 significant digits with η. It appears that the computational convergence order well approaches 8. Table 4. Convergence for test functions F1 (x) − F3 (x) with typically selected methods AX2, BX1, DY7. MT

F

WAX2

F1

WBX1

F2

n

xn

|F(xn )|

|xn − α|

0 1 2 3 0 1 2 3

1.7 1.81648572902243 1.81648572902222 1.81648572902222 1.87 1.99999999393956 2.00000000000000 2.00000000000000 ∗ 1.97 −0.36

0.525256 9.564 × 10−13 4.601 × 10−108 2.425 × 10−173 1.62893 8.327 × 10−8 9.980 × 10−67 8.086 × 10−174

0.116486 2.091 × 10−13 1.006 × 10−108 4.043 × 10−174 0.130000 6.060 × 10−9 7.262 × 10−68 2.425 × 10−173

0.0608640

0.0789358

0

η

pn

6.169498388 × 10−6 2.749110983 × 10−7

2.749110983 × 10−7

6.55305 8.00000

0.07429458432 0.03990968332

0.0399096822

7.69542 8.00000

1.326 × 10−8

8.801529463

4.234524089

7.71185

4.062 × 10−63

4.234524636

1.99999999638476 1.148 × 10−8 −0.433012689128059 2.00000000000000 3.518 × 10−63 WDY7 F3 2 −0.433012701892219 2.00000000000000 3 0.0 × 10−160 −0.433012701892219 ∗ √ 1.97 Note: MT = method, −0.36 = 1.96 − 0.36i, i = −1. 1

|en /e8n−1 |

5.717 × 10−174

8.00000

26

M. S. RHEE ET AL.

Table 5. Additional test functions fi (x) with zeros α and initial guesses x0 . fi (x)

i 1 2

cos [(x

x sin (x 2 ) − log [1 + x12 − π1 ] − 3)2 + 3] − log [(x − 3)2 + 4] − 1

α √ π √ 3+i 3

2.95 + 1.76i

x0 1.7

3

x 5 + log [1 + sin x]

0

0.05

4

sin−1 ( 1x − 1) − 4x 2 + 3 √ x 3 − π 3 + sin x (x + 1)

0.884690687180673

1.0

π

2.9

4x 2 + e−x + sin (1 + 1x ) − 4 √ log x − x + x 3

0.830382156106894

0.75

1

0.9

5 6 7


Note: Here, log z (z ∈ C) representsaprincipalanalyticbranchwith − π ≤ Im(log z) < π .

Table 5 lists additional test functions to ensure the convergence behaviour of proposed scheme (2). In Table 6, we compare numerical errors |xn − α| of proposed methods WAX1 through WFY6 with that of method SA. The least errors within the prescribed error bound are highlighted in bold face. Although we are limited to the selected current experiments, within two iterations, a strict comparison shows that methods WFY6, WDY7, WCY1, WDX2 display slightly better convergence for test functions f1 , f2 , f5 , f6 , respectively, and method WFY4 for three test functions f3 , f4 , f7 . In view of a close inspection of the asymptotic error constant η(θi , Lf , Kf ) = |xn+1 − α|/|xn − α|8 , we should be aware that the local convergence is dependent on the function f (x), an initial value x0 , the zero α itself and the weight functions Lf and Kf . Accordingly, for all given set of test functions, the convergence of one method is hardly expected to be always better than the others. The efficiency index EI [40] is found to be 81/4 ≈ 1.68179 for the proposed family of methods (2), which evidently show a better performance than that of classical Newton’s method. Proper initial values generally influence the convergence behaviour of iterative methods. To guarantee the convergence of Newton-like iterative map (54) with a weight function Hp (z), it requires good initial values close to zero α. It is, however, not a simple task to determine how close the initial values are to zero α, since initial values are generally sensitive to computational precision, error bound and the given function f (x) under consideration. We now introduce the notion of the basin of attraction that is the set of initial guesses leading to long-time behaviour approaching the attractors (e.g. periodic, quasi-periodic or chaotic behaviours of different types) under the action of the iterative function. Hence, one effective way of selecting stable initial values would be directly using visual basins of attraction. Since the area of convergence can be seen on the basins of attraction, it would be reasonable to say that a method having a larger area of convergence implies a more robust method. It is no doubt for us to employ a quantitative analysis for measuring the size of area of convergence. Evidently, convergence behaviour of global character can be conveniently observed on the basin of attraction. The basic topological structure of such a basin of attraction as a region can vary greatly from system to system with various forms of weight functions. To show the performance of the listed methods, we present Tables 7–9 featuring a statistical data giving the average number of iterations per point, CPU time (in seconds) and number of points requiring 40 iterations. In the following examples, we take a 6 × 6 square centred at the origin and containing all the zeros of the given functions. We then take 601 × 601 equally spaced points in the square as initial points for the iterative methods. We colour the point based on the root it converged to. This way we can figure out if the method converged within the maximum number of iteration allowed and if it converged to the root closer to the initial point.


27

Table 6. Comparison of |xn − α| for selected methods applied to various test functions. f (x); x0 Method

|xn − α|

f1 ; 1.7

f2 ; 2.95 + 1.76i

f3 ; 0.05

f4 ; 1.0

f5 ; 2.9

f6 ; 0.75

f7 ; 0.9

WAX1

|x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α| |x1 − α| |x2 − α|

1.35e−8∗ 7.13e−63 2.57e−8 2.88e−60 1.87e−8 2.37e−61 2.79e−8 5.19e−60 3.08e−8 8.46e−60 3.20e−8 1.27e−59 1.57e−8 3.88e−62 2.60e−8 4.31e−60 5.44e−8 2.93e−57 7.75e−9 2.36e−65 1.15e−8 1.27e−63 9.61e−9 5.87e−64 3.52e−8 3.88e−59 1.93e−8 2.74e−61 5.49e−8 2.73e−57 2.05e−8 1.35e−61 2.01e−7 2.26e−52 8.60e−9 1.66e−64 4.29e−8 3.01e−58 2.20e−8 9.47e−61 2.61e−8 1.25e−60 1.79e−8 1.88e−61 3.15e−8 2.49e−59 1.45e−8 2.01e−62 4.56e−9 9.29e−67 2.66e−8 5.10e−60

4.21e−9 5.71e−67 7.63e−9 1.50e−64 3.09e−9 7.13e−68 6.17e−9 2.12e−65 4.73e−9 1.75e−66 3.29e−9 5.49e−68 7.86e−9 1.92e−64 5.22e−9 7.23e−66 9.17e−9 1.07e−63 1.28e−9 3.40e−71 3.57e−9 1.15e−67 1.67e−9 7.31e−71 6.10e−9 1.89e−65 1.76e−9 3.68e−70 1.41e−8 4.35e−62 5.45e−10 1.50e−74 2.37e−8 4.77e−60 4.96e−9 2.70e−66 1.17e−8 8.11e−63 9.82e−9 1.52e−63 5.84e−9 1.14e−65 5.28e−9 8.08e−66 1.38e−8 3.66e−62 7.85e−9 1.93e−64 1.98e−9 4.73e−71 5.43e−9 9.89e−66





3.56e−10 8.88e−78 4.18e−10 3.88e−77 2.92e−10 1.54e−78 3.17e−10 2.96e−78 2.21e−10 9.39e−80 1.30e−10 3.11e−82 7.11e−10 4.79e−75 2.80e−10 1.01e−78 2.20e−10 1.24e−79 3.50e−10 6.90e−78 2.36e−10 2.03e−79 1.97e−10 4.21e−80 2.80e−10 8.74e−79 1.79e−10 1.80e−80 6.92e−10 3.53e−75 3.76e−10 1.27e−77 3.79e−10 1.13e−77 5.26-10 3.19e−76 6.31e−10 1.57e−75 5.80e−10 7.76e−76 7.68e−10 1.06e−74 4.23e−10 4.27e−77 7.64e−11 3.22e−84 1.39e−10 1.52e−81 2.07e−10 6.32e−80 2.65e−10 6.58e−79

WAX2 WAX6 WAY4 WAY5 WAY6 WBX1


WBX6 WBY6 WBZ5 WCX4 WCY1 WCY2 WCY6 WDX2 WDY7 WDY9 WEX2 WEX6 WEY6 WEZ1 WEZ7 WFY4 WFY5 WFY6 SA ∗

1.35e − 8 ≡ 1.35 × 10−8 .

Although we have experimented with all methods listed in Table 2, because of space limitations, we are only able to present the dynamics of 20 selected iterative maps WAX1, WAX6, WAY4, WAY5, WAY6, WBX1, WBX6, WBY6, WBZ5, WCX4, WCY2, WCY6, WDY9, WEX6, WEY6, WEZ1, WEZ7, WFY4, WFY6 and SA, when applied to various polynomials pk (z), (1 ≤ k ≤ 6) and one non-polynomial equation.

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Example 5.1: As a first example, we have taken a quadratic polynomial with all real roots: p1 (z) = z2 − 1.

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Clearly, the roots are ±1. Basins of attraction for WAX1–WFY6 and SA are given in Figure 1. Consulting Tables 7–9, we find that the method SA uses the least number of iterations per point on average (ANIP), it also has the least number of black points. The methods WAX6, WCY6 and WEZ7 have almost the same ANIP as SA. The fastest method is SA with 153.130 s.

Table 7. Average number of iterations per point for each example (1–7). Example


Map WAX1 WAX6 WAY4 WAY5 WAY6 WBX1 WBX6 WBY6 WBZ5 WCX4 WCY2 WCY6 WDY9 WEX6 WEY6 WEZ1 WEZ7 WFY4 WFY6 SA

1

2

3

4

5

6

7

Average

2.25 2.18 2.31 2.35 2.42 2.22 2.22 2.28 2.28 2.28 2.39 2.18 2.28 2.28 2.33 2.25 2.18 2.40 2.24 2.16

2.52 2.46 2.62 2.74 2.90 2.54 2.58 2.73 2.63 2.60 2.77 2.47 2.57 2.51 2.59 2.51 2.51 2.66 2.57 2.46

2.68 2.64 2.81 2.95 3.11 2.71 2.79 2.72 2.81 2.78 2.99 2.70 2.76 2.70 2.75 2.65 2.67 2.91 2.76 2.62

3.09 3.11 3.10 3.20 3.59 3.45 3.40 3.77 3.40 3.05 3.16 3.51 3.72 3.11 5.83 3.74 3.38 3.31 3.27 2.98

3.11 3.30 3.06 3.21 3.65 4.55 3.52 4.07 3.67 3.07 3.15 3.72 5.08 3.12 6.24 4.14 3.71 3.52 3.33 3.03

2.96 3.13 3.02 3.15 3.46 4.02 3.33 3.94 3.16 2.96 3.08 3.60 3.96 3.28 5.12 3.42 3.11 3.49 3.22 2.89

2.24 2.29 2.29 2.43 2.45 2.47 2.44 2.69 2.46 2.34 2.37 2.59 2.63 2.47 2.49 2.40 2.25 2.80 2.41 2.28

2.69 2.73 2.75 2.86 3.08 3.14 2.90 3.17 2.92 2.73 2.84 2.97 3.29 2.78 3.91 3.01 2.83 3.01 2.83 2.63

Table 8. CPU time (in seconds) required for each example (1–7) using a Dell Multiplex-990. Example Map WAX1 WAX6 WAY4 WAY5 WAY6 WBX1 WBX6 WBY6 WBZ5 WCX4 WCY2 WCY6 WDY9 WEX6 WEY6 WEZ1 WEZ7 WFY4 WFY6 SA

1

2

3

4

5

6

7

Average

201.116 192.630 179.417 197.762 195.048 182.147 187.731 200.758 184.299 169.963 192.864 188.402 190.259 170.961 199.572 186.093 191.476 205.844 194.471 153.130

313.562 313.999 314.139 336.588 339.692 304.560 310.598 347.227 323.187 289.834 327.103 310.535 314.139 279.725 317.742 306.776 317.025 336.323 315.262 290.364

292.050 274.749 277.307 310.629 304.733 260.678 274.998 280.240 271.972 256.092 283.719 274.889 280.006 246.919 277.192 265.544 284.655 300.271 292.767 241.786

340.425 348.117 331.190 358.272 377.148 343.592 352.905 415.446 351.891 297.104 334.545 370.580 392.217 307.884 637.857 396.368 368.271 363.014 359.067 294.529

399.175 418.722 362.250 391.766 431.000 520.326 426.398 499.999 446.911 350.222 382.624 453.776 592.211 344.216 749.054 486.208 449.158 437.208 405.041 338.740

983.539 1018.156 956.364 1019.825 1091.351 1258.648 1062.882 1270.940 1018.858 936.537 992.900 1168.276 1278.116 1024.989 1651.052 1099.385 1003.742 1135.063 1037.720 1252.002

359.536 354.404 336.041 363.654 358.240 367.569 354.934 409.627 368.599 326.572 356.353 375.027 391.578 354.075 375.915 351.549 337.024 419.066 348.943 333.967

412.772 417.253 393.815 425.499 442.459 462.503 424.349 489.177 423.674 375.189 410.015 448.784 491.218 389.824 601.198 441.703 421.622 456.684 421.896 414.931


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Table 9. Number of points requiring 40 iterations for each example (1–7). Example


Map WAX1 WAX6 WAY4 WAY5 WAY6 WBX1 WBX6 WBY6 WBZ5 WCX4 WCY2 WCY6 WDY9 WEX6 WEY6 WEZ1 WEZ7 WFY4 WFY6 SA

1

2

3

4

5

6

7

Average

747 769 795 949 1175 753 749 761 769 765 1433 769 761 765 1281 741 773 1021 745 601

50 57 117 534 1217 46 15 19 109 70 1162 11 53 23 549 84 83 14 10 54

0 0 0 476 1024 0 0 0 0 0 1260 0 0 0 16 0 0 0 0 0

1201 1345 1201 1201 2001 1201 1201 1201 1337 1201 1201 1201 1261 1201 28,661 1433 1257 1201 1201 1201

1 1451 5 6 959 1 1 19 1619 1 1 6 843 1 32,016 1879 1450 200 1 1

0 1086 0 158 1027 0 5 23 5 0 1 10 0 0 16,657 2 2 5 2 0

330 694 914 842 1314 533 1714 1884 1580 556 972 1564 1598 1141 1191 950 705 2395 1212 514

333 772 433 595 1245 362 526 558 774 370 861 509 645 447 11,482 727 610 691 453 339

Example 5.2: In our second example, we have taken a cubic polynomial: p2 (z) = z3 + 4z2 − 10.

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Basins of attraction are given in Figure 2. We now consult the tables to find that the method with the fewest ANIP are SA and WAX6 with 2.46 iteration. All the others require between 2.51 and 2.90. In terms of CPU time in seconds, the fastest is WEX6 (279.725 s) and the slowest is WBY6 (347.227 s). The method WAY6 has the most black points (1216) and WFY6 has the least (10 points). Example 5.3: As a third example, we have taken another cubic polynomial: p3 (z) = z3 − z.

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Now all the roots are real. The basins for this example are plotted in Figure 3. Based on Table 7, we see that again SA has the lowest ANIP followed closely by WAX6. The fastest method is again SA (241.786 s) followed by WEX6 (246.919 s) and the slowest are WAY5 (310.629 s) and WAY6 (304.733 s). Most of the methods have no black points except WCY2 with 1260, WAY6 with 1024, WAY5 with 476 and WEY6 with 16 black points. Example 5.4: As a fourth example, we have taken a quartic polynomial: p4 (z) = z4 − 1.

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The basins are given in Figure 4. We now see that WBZ5, WDY9, WEY6, WEZ1 and WEZ7 are the worst. The best are those with smaller lobes along the diagonals. In terms of ANIP, SA is the best (2.98) and WEY6 is the worst (5.83). The fastest is again SA (294.529 s) followed by WCX4 (297.104 s) and the slowest is WEY6 (637.857 s). Most of the methods have 1201 black point with the worst being WEY6 with 28,661 points.

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Figure 1. The top row for WAX1 (left), WAX6 (centre left), WAY4 (centre right) and WAY5 (right). The second row for WAY6 (left), WBX1 (centre left), WBX6 (centre right) and WBY6 (right). The third row for WBZ5 (left), WCX4 (centre left), WCY2 (centre right) and WCY6 (right). The fourth row for WDY9 (left), WEX6 (centre left), WEY6 (centre right) and WEZ1 (right). The bottom row for WEZ7 (left), WFY4 (centre left), WFY6 (centre right) and SA (right), for the roots of the polynomial (z2 − 1).

Example 5.5: As a fifth example, we have taken a quintic polynomial: p5 (z) = z5 − 1.

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The basins for the best methods left are plotted in Figure 5. The worst are WDY9, WEY6, WEZ1, WEZ7, WBZ5 and WBX1. In terms of ANIP, the best is SA (3.03) followed closely by WAY4 (3.06) and WCX4 (3.07) and the worst are WEY6 (6.24) and WDY9 (5.08). The fastest is SA using 338.74 s followed by WEX6 using 344.216 s and the slowest is WEY6 (749.054 s). There are 16 methods with less than 10 black points. The highest number is for WEY6 (32,016) preceded by WEZ1 with 1879 black points.

31



Figure 2. The top row for WAX1 (left), WAX6 (centre left), WAY4 (centre right) and WAY5 (right). The second row for WAY6 (left), WBX1 (centre left), WBX6 (centre right) and WBY6 (right). The third row for WBZ5 (left), WCX4 (centre left), WCY2 (centre right) and WCY6 (right). The fourth row for WDY9 (left), WEX6 (centre left), WEY6 (centre right) and WEZ1 (right). The bottom row for WEZ7 (left), WFY4 (centre left), WFY6 (centre right) and SA (right), for the roots of the polynomial (z3 + 4z2 − 10).

Example 5.6: As a sixth example, we have taken a sextic polynomial with complex coefficients: 1 11(i + 1) 4 3i + 19 3 5i + 11 2 i + 11 3 z − z + z − z + −3i. p6 (z) = z6 − z5 + 2 4 4 4 4 2

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The basins for the best methods left are plotted in Figure 6. It seems that the best methods are WAX6, WAY6, WBX6, WBY6, WCY6 and WFY6. The worst are WBX1 and WDY9. Based on Table 7, we find that SA has the lowest ANIP (2.89) followed by WAX1 and WCX4 (2.96). The fastest method is WCX4 (936.537s) followed by WAY4 (956.364s) and WAX1 (983.539s). There are 10 methods

M. S. RHEE ET AL.


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Figure 3. The top row for WAX1 (left), WAX6 (centre left), WAY4 (centre right) and WAY5 (right). The second row for WAY6 (left), WBX1 (centre left), WBX6 (centre right) and WBY6 (right). The third row for WBZ5 (left), WCX4 (centre left), WCY2 (centre right) and WCY6 (right). The fourth row for WDY9 (left), WEX6 (centre left), WEY6 (centre right) and WEZ1 (right). The bottom row for WEZ7 (left), WFY4 (centre left), WFY6 (centre right) and SA (right), for the roots of the polynomial (z3 − z).

without black points and 10 methods with 10 or less. The highest number is for WEY6 with 16,657 black points. Example 5.7: As a last example, we have taken a non-polynomial equation: p7 (z) = (ez+1 − 1)(z + 1).

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The basins for this example are plotted in Figure 7. the roots are at ±1 and it is expected that the boundary will be close to the imaginary axis as in Example 1. All methods show a larger basin for the

33



Figure 4. The top row for WAX1 (left), WAX6 (centre left), WAY4 (centre right) and WAY5 (right). The second row for WAY6 (left), WBX1 (centre left), WBX6 (centre right) and WBY6 (right). The third row for WBZ5 (left), WCX4 (centre left), WCY2 (centre right) and WCY6 (right). The fourth row for WDY9 (left), WEX6 (centre left), WEY6 (centre right) and WEZ1 (right). The bottom row for WEZ7 (left), WFY4 (centre left), WFY6 (centre right) and SA (right), for the roots of the polynomial for the roots of the polynomial (z4 − 1).

root at −1. The methods with the largest basin for +1 are WAX1, WAY4, WCY2, WEY6 and WEZ1. In terms of ANIP, WAX1 is best (2.24) followed closely by WEZ7 (2.25), SA (2.28) and WAX6, and WAY4 with 2.29. The worst is WFY4 with 2.80. The fastest method is WCX4 (326.572s) and the slowest is WFY4 (419.066s). WAX1 has the least number of black points and WFY4 has the highest (2395) such number. Based on these seven examples we see that SA has six examples with the lowest ANIP, WAX1 and WAX6 each with one example. WCX4 is the fastest in two examples, WEX6 in one example and SA is the fastest in the other four examples.

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34

Figure 5. The top row for WAX1 (left), WAX6 (centre left), WAY4 (centre right) and WAY5 (right). The second row for WAY6 (left), WBX1 (centre left), WBX6 (centre right) and WBY6 (right). The third row for WBZ5 (left), WCX4 (centre left), WCY2 (centre right) and WCY6 (right). The fourth row for WDY9 (left), WEX6 (centre left), WEY6 (centre right) and WEZ1 (right). The bottom row for WEZ7 (left), WFY4 (centre left), WFY6 (centre right) and SA (right), for the roots of the polynomial (z5 − 1).

We now average all these results across the seven examples to try and pick the best method. SA has the lowest ANIP (2.63) followed by WAX1 with 2.69, WAX6 and WCX4 with 2.73. The fastest method is WCX4 followed by WEX6 (389.824s). WAX1 has the lowest number of black points on average (333) followed by SA, WBX1 and WCX4. Based on this, we recommend WCX4 since it is the only method mentioned as close to the top at all three categories. SA and WAX1 are close to the top at 2 out of the three categories. As concluding remarks of our study, we state the following results. Theorem 2.1 verifies that convergence order of proposed family of methods (2) has been increased to 8 by means of weight

35



Figure 6. The top row for WAX1 (left), WAX6 (centre left), WAY4 (centre right) and WAY5 (right). The second row for WAY6 (left), WBX1 (centre left), WBX6 (centre right) and WBY6 (right). The third row for WBZ5 (left), WCX4 (centre left), WCY2 (centre right) and WCY6 (right). The fourth row for WDY9 (left), WEX6 (centre left), WEY6 (centre right) and WEZ1 (right). The bottom row for WEZ7 (left), WFY4 (centre left), WFY6 (centre right) and SA (right), for the roots of the polynomial z6 − 12 z5 + 11(i + 1)/4z4 − ((3i + 19)/4)z3 + ((5i + 11)/4)z2 − ((i + 11)/4)z + 32 − 3i.

functions dependent upon function-to-function ratios in their second and third sub-steps. Computational aspects through a variety of test equations for selected cases well agree with the developed theory, verifying the convergence order as well as asymptotic error constants. Dynamical aspects among listed methods have been also illustrated through their basins of attraction not only with a qualitative stability analysis on purely imaginary extraneous fixed points for a prototype quadratic polynomial f (z) = z2 − 1 motivated by the earlier work of Vrscay and Gilbert [42], but also with a quantitative statistical analysis for various polynomials pk (z) as well as a non-polynomial example.

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36

Figure 7. The top row for WAX1 (left), WAX6 (centre left), WAY4 (centre right) and WAY5 (right). The second row for WAY6 (left), WBX1 (centre left), WBX6 (centre right) and WBY6 (right). The third row for WBZ5 (left), WCX4 (centre left), WCY2 (centre right) and WCY6 (right). The fourth row for WDY9 (left), WEX6 (centre left), WEY6 (centre right) and WEZ1 (right). The bottom row for WEZ7 (left), WFY4 (centre left), WFY6 (centre right) and SA (right), for the roots of the non-polynomial equation (ez+1 − 1)(z + 1).

We can determine which members of the proposed family of methods (2) give better convergence from the illustrative basins of attraction. In our future study, we will extend the current approach with other types of weight functions by means of a different selection of parameters to a high-order family of simple- or multiple-root finders in order to enhance the desired dynamical characteristics behind their purely imaginary extraneous fixed points.

Disclosure statement No potential conflict of interest was reported by the authors.


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An optimal eighth-order class of three-step weighted

An optimal eighth-order class of three-step weighted

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